1 00:00:00,500 --> 00:00:06,560 In this video, we're going to be looking at sequences and series, 2 00:00:06,560 --> 00:00:11,105 so let's begin by looking at what a sequences. 3 00:00:12,030 --> 00:00:15,885 This, for instance is a 4 00:00:15,885 --> 00:00:23,280 sequence. It's a set of numbers. 5 00:00:23,830 --> 00:00:30,745 And here we seem to have a rule. All of these are odd numbers, or 6 00:00:30,745 --> 00:00:36,277 we can look at it. We increase by two each time 13579. 7 00:00:37,280 --> 00:00:42,460 So there's our sequence of odd numbers. 8 00:00:43,900 --> 00:00:45,788 Here is another sequence. 9 00:00:46,650 --> 00:00:53,742 These numbers are the 10 00:00:53,742 --> 00:01:01,102 square numbers. 1 squared, 2 squared, is 4 three 11 00:01:01,102 --> 00:01:07,066 squared is 9. Four squared is 16 and 5 squared is 25. So again 12 00:01:07,066 --> 00:01:12,604 we've got a sequence of numbers. We've got a rule that seems to 13 00:01:12,604 --> 00:01:15,269 produce them. Those are the. 14 00:01:15,890 --> 00:01:22,140 Square numbers. Is a slightly different sequence. 15 00:01:22,510 --> 00:01:27,450 Here we've got alternation between one and minus one back 16 00:01:27,450 --> 00:01:34,366 to one on again to minus one back to one on again to minus 17 00:01:34,366 --> 00:01:38,812 one. But this is still a sequence of numbers. 18 00:01:39,510 --> 00:01:42,966 Now, because I've written some dots after it here. 19 00:01:43,660 --> 00:01:48,769 This means that this is meant to be an infinite sequence. It goes 20 00:01:48,769 --> 00:01:53,485 on forever and this is meant to be an infinite sequence. It 21 00:01:53,485 --> 00:01:58,594 carries on forever, and this one does too. If I want a finite 22 00:01:58,594 --> 00:02:05,570 sequence. What might a finite sequence 23 00:02:05,570 --> 00:02:09,188 look like, for 24 00:02:09,188 --> 00:02:15,904 instance 1359? That would be a finite sequence. We've got 25 00:02:15,904 --> 00:02:18,652 4 numbers and then it stops 26 00:02:18,652 --> 00:02:25,160 dead. Perhaps if we look at the sequence of square 27 00:02:25,160 --> 00:02:30,960 numbers 1, four, 916, that again is a finite sequence. 28 00:02:32,070 --> 00:02:39,950 Sequence that will be very interested in is the sequence 29 00:02:39,950 --> 00:02:47,042 of whole numbers, the counting numbers, the integers. So 30 00:02:47,042 --> 00:02:54,922 there's a sequence of integers, and it's finite because it 31 00:02:54,922 --> 00:03:02,802 stops at N, so we're counting 123456789 up to N. 32 00:03:02,820 --> 00:03:08,150 And the length of this sequence is an the integer, the number N. 33 00:03:10,310 --> 00:03:16,277 Very popular sequence of numbers. Quite well known is 34 00:03:16,277 --> 00:03:18,266 this particular sequence. 35 00:03:19,000 --> 00:03:23,610 This is a slightly different sequence. It's infinite keeps on 36 00:03:23,610 --> 00:03:25,915 going and it's called the 37 00:03:25,915 --> 00:03:30,714 Fibonacci sequence. And we can see how it's generated. This 38 00:03:30,714 --> 00:03:35,823 number 2 is formed by adding the one and the one together, and 39 00:03:35,823 --> 00:03:40,539 then the three is formed by adding the one and the two 40 00:03:40,539 --> 00:03:45,255 together. The Five is formed by adding the two and the three 41 00:03:45,255 --> 00:03:46,827 together, and so on. 42 00:03:47,840 --> 00:03:52,585 So there's a question here. How could we write this rule down in 43 00:03:52,585 --> 00:03:56,600 general, where we can say it that any particular term is 44 00:03:56,600 --> 00:04:00,615 generated by adding the two numbers that come before it in 45 00:04:00,615 --> 00:04:04,630 the sequence together? But how might we set that down? How 46 00:04:04,630 --> 00:04:09,740 might we label it? One way might be to use algebra and say will 47 00:04:09,740 --> 00:04:14,120 call the first term you, and because it's the first time we 48 00:04:14,120 --> 00:04:17,040 want to label it, so we call it 49 00:04:17,040 --> 00:04:20,690 you one. And then the next terminal sequence, the second 50 00:04:20,690 --> 00:04:25,305 term. It would make sense to call it you too, and the third 51 00:04:25,305 --> 00:04:29,565 term in our sequence. It would make sense therefore, to call it 52 00:04:29,565 --> 00:04:36,916 you 3. You four and so on up to UN. So this represents a 53 00:04:36,916 --> 00:04:43,012 finite sequence that's got N terms in it. If we look at 54 00:04:43,012 --> 00:04:48,600 the Fibonacci sequence as an example of making use of this 55 00:04:48,600 --> 00:04:54,696 kind of notation, we could say that the end term UN was 56 00:04:54,696 --> 00:04:59,776 generated by adding together the two terms that come immediately 57 00:04:59,776 --> 00:05:01,300 before it will. 58 00:05:01,320 --> 00:05:05,110 Term that comes immediately before this must have a number 59 00:05:05,110 --> 00:05:10,416 attached to it. That's one less than N and that would be N minus 60 00:05:10,416 --> 00:05:15,910 one. Plus on the term that's down, the term that comes 61 00:05:15,910 --> 00:05:20,926 immediately before this one must have a number attached to it. 62 00:05:20,926 --> 00:05:26,854 That's one less than that. Well, that's UN minus 1 - 1 taking 63 00:05:26,854 --> 00:05:30,046 away 2 ones were taking away two 64 00:05:30,046 --> 00:05:36,642 altogether. So that we can see how we might use the algebra 65 00:05:36,642 --> 00:05:42,546 this algebraic notation help us write down a rule for the Fibo 66 00:05:42,546 --> 00:05:46,406 Nachi sequence. OK, how can 67 00:05:46,406 --> 00:05:52,185 we? Use this in a slightly different way. 68 00:05:52,710 --> 00:06:00,270 What we need to look at now is to move on and have a 69 00:06:00,270 --> 00:06:04,050 look what we mean by a series. 70 00:06:04,630 --> 00:06:11,716 This is a sequence, label it. 71 00:06:12,490 --> 00:06:17,330 A sequence it's a list of numbers generated by some 72 00:06:17,330 --> 00:06:22,170 particular rule. It's finite because there are any of them. 73 00:06:22,900 --> 00:06:28,510 What then, is a series series is what we get. 74 00:06:29,280 --> 00:06:32,970 When we add. 75 00:06:32,970 --> 00:06:36,478 Terms of the sequence. 76 00:06:37,070 --> 00:06:44,078 Together And because we're adding together and terms 77 00:06:44,078 --> 00:06:46,934 will call this SN. 78 00:06:47,860 --> 00:06:55,610 The sum of N terms and it's that which is 79 00:06:55,610 --> 00:06:57,160 the series. 80 00:06:59,040 --> 00:07:06,097 So. Let's have a look at the sequence 81 00:07:06,097 --> 00:07:12,825 of numbers 123456, and so on up to 82 00:07:12,825 --> 00:07:13,666 N. 83 00:07:15,080 --> 00:07:17,190 Then S1. 84 00:07:18,850 --> 00:07:22,210 Is just one. 85 00:07:22,840 --> 00:07:24,700 S2. 86 00:07:25,750 --> 00:07:32,660 Is the sum of the first 2 terms 1 + 87 00:07:32,660 --> 00:07:34,980 2? And that gives us 3. 88 00:07:35,740 --> 00:07:42,893 S3. Is the sum of the first three 89 00:07:42,893 --> 00:07:46,607 terms 1 + 2 + 3? 90 00:07:47,220 --> 00:07:50,805 And that gives us 6 91 00:07:50,805 --> 00:07:58,250 and S4. Is the sum of the first four terms 1 + 92 00:07:58,250 --> 00:08:02,100 2 + 3 + 4 and that 93 00:08:02,100 --> 00:08:03,580 gives us. Hey. 94 00:08:04,480 --> 00:08:10,148 So this gives us the basic vocabulary to be able to move on 95 00:08:10,148 --> 00:08:14,944 to the next section of the video, but just let's remind 96 00:08:14,944 --> 00:08:16,688 ourselves first of all. 97 00:08:17,220 --> 00:08:18,640 A sequence. 98 00:08:20,180 --> 00:08:24,203 Is a set of numbers generated by some rule. 99 00:08:25,830 --> 00:08:31,402 A series is what we get when we add the terms of the sequence 100 00:08:31,402 --> 00:08:37,516 together. This particular sequence has N terms in it 101 00:08:37,516 --> 00:08:43,486 because we've labeled each term in the sequence with accounting 102 00:08:43,486 --> 00:08:49,456 number. If you like U1U2U free, you fall you N. 103 00:08:51,380 --> 00:08:55,921 Now. With this vocabulary of sequences and series in mind, 104 00:08:55,921 --> 00:09:01,756 we're going to go on and have a look at a 2 special kinds of 105 00:09:01,756 --> 00:09:06,630 sequences. The first one is called an arithmetic progression 106 00:09:06,630 --> 00:09:09,990 and the second one is called a 107 00:09:09,990 --> 00:09:15,125 geometric progression. Will begin with an arithmetic 108 00:09:15,125 --> 00:09:22,062 progression. Let's start by having a look at this 109 00:09:22,062 --> 00:09:24,950 sequence of. Odd 110 00:09:25,470 --> 00:09:32,748 Numbers that we had before 1357. 111 00:09:33,300 --> 00:09:40,026 Is another sequence not 1020 thirty, 112 00:09:40,026 --> 00:09:43,389 and so on. 113 00:09:44,640 --> 00:09:51,634 What we can see in this first sequence is that each term after 114 00:09:51,634 --> 00:09:56,476 the first one is formed by adding on to. 115 00:09:57,210 --> 00:10:00,258 1 + 2 gives us 3. 116 00:10:00,840 --> 00:10:03,858 3 + 2 gives us 5. 117 00:10:04,710 --> 00:10:11,145 5 + 2 gives us 7 and it's because we're adding on the 118 00:10:11,145 --> 00:10:17,085 same amount every time. This is an example of what we call 119 00:10:17,085 --> 00:10:18,075 an arithmetic. 120 00:10:19,870 --> 00:10:28,082 Progression. If we look at this sequence of 121 00:10:28,082 --> 00:10:33,538 numbers, we can see exactly the same property we've started with 122 00:10:33,538 --> 00:10:39,986 zero. We've added on 10, and we've added on 10 again to get 123 00:10:39,986 --> 00:10:46,930 20. We've had it on 10 again to get 30, so again, this is 124 00:10:46,930 --> 00:10:49,410 exactly the same. It's an 125 00:10:49,410 --> 00:10:56,166 arithmetic progression. We don't have to add on things, so 126 00:10:56,166 --> 00:11:03,283 for instance a sequence of numbers that went like this 8 127 00:11:03,283 --> 00:11:05,871 five, 2 - 1. 128 00:11:06,560 --> 00:11:08,665 Minus 129 00:11:08,665 --> 00:11:14,253 4. If we look what's happening where going from 8:00 130 00:11:14,253 --> 00:11:18,510 to 5:00, so that's takeaway three were going from five to 131 00:11:18,510 --> 00:11:23,154 two, so that's takeaway. Three were going from 2 to minus. One 132 00:11:23,154 --> 00:11:27,411 takeaway. Three were going from minus one to minus four takeaway 133 00:11:27,411 --> 00:11:33,662 3. Another way of thinking about takeaway three is to say where 134 00:11:33,662 --> 00:11:35,510 adding on minus three. 135 00:11:36,330 --> 00:11:43,050 8 at minus three is 5 five at minus three is 2, two AD minus 136 00:11:43,050 --> 00:11:48,426 three is minus one, so again, this is an example of an 137 00:11:48,426 --> 00:11:54,084 arithmetic progression. And what we want to be able to do is to 138 00:11:54,084 --> 00:11:56,996 try and encapsulate this arithmetic progression in some 139 00:11:56,996 --> 00:11:59,180 algebra, so we'll use the letter 140 00:11:59,180 --> 00:12:04,686 A. To stand for the first term. 141 00:12:05,310 --> 00:12:12,360 And will use the letter D to stand for the 142 00:12:12,360 --> 00:12:17,768 common difference. Now the common difference is the 143 00:12:17,768 --> 00:12:23,686 difference between each term and it's called common because it is 144 00:12:23,686 --> 00:12:30,142 common to each between each term. So let's have a look at 145 00:12:30,142 --> 00:12:36,598 one 357 and let's have a think about how it's structured 13. 146 00:12:37,340 --> 00:12:44,820 5. 7 and so on. So we begin with one and 147 00:12:44,820 --> 00:12:48,817 then the three is 1 + 2. 148 00:12:50,270 --> 00:12:57,755 The Five is 1 + 2 tools because by the time we got to five, 149 00:12:57,755 --> 00:13:00,749 we've added four onto the one. 150 00:13:01,420 --> 00:13:07,894 The Seven is one plus. Now the time we've got to Seven, we've 151 00:13:07,894 --> 00:13:14,368 added three tools on. Let's just do one more. Let's put nine in 152 00:13:14,368 --> 00:13:19,846 there and that would be 1 + 4 times by two. 153 00:13:20,980 --> 00:13:25,320 So let's see if we can begin to write this down. This is one. 154 00:13:26,280 --> 00:13:33,235 Now what have we got here? This is the second term in the 155 00:13:33,235 --> 00:13:39,773 series. But we've only got 1 two there, so if you like we've got 156 00:13:39,773 --> 00:13:43,141 1 + 2 - 1 times by two. 157 00:13:44,060 --> 00:13:48,776 One plus now, what's multiplying the two here? Well, this is the 158 00:13:48,776 --> 00:13:50,741 third term in the series. 159 00:13:51,650 --> 00:13:59,306 So we've got a 2 here, so we're multiplying by 3 - 160 00:13:59,306 --> 00:14:05,199 1. Here this is term #4 and we're 161 00:14:05,199 --> 00:14:10,932 multiplying by three, so that's 4 - 1 times 162 00:14:10,932 --> 00:14:17,302 by two. And here this is term #5, so we've 163 00:14:17,302 --> 00:14:23,035 got 1 + 5 - 1 times by two. 164 00:14:24,430 --> 00:14:28,670 Now, if we think about what's happening here. 165 00:14:31,120 --> 00:14:33,268 We're starting with A. 166 00:14:34,670 --> 00:14:37,610 And then on to the A. We're 167 00:14:37,610 --> 00:14:44,940 adding D. Then we're adding on another day, so that's a plus 168 00:14:44,940 --> 00:14:51,828 2D, and then we're adding on another D. So that's a plus 169 00:14:51,828 --> 00:14:57,214 3D. The question is, if we've got N terms in our sequence, 170 00:14:57,214 --> 00:15:02,450 then what's the last term? But if we look, we can see that the 171 00:15:02,450 --> 00:15:04,320 first term was just a. 172 00:15:04,880 --> 00:15:12,244 The second term was a plus, one D. The third term was a plus 173 00:15:12,244 --> 00:15:19,608 2D. The fourth term was a plus 3D, so the end term must be 174 00:15:19,608 --> 00:15:22,238 a plus N minus one. 175 00:15:22,750 --> 00:15:23,230 Gay. 176 00:15:25,200 --> 00:15:32,150 Now, this last term of our sequence, we often label 177 00:15:32,150 --> 00:15:35,625 L and call it the 178 00:15:35,625 --> 00:15:37,015 last term. 179 00:15:37,070 --> 00:15:39,650 Or 180 00:15:40,420 --> 00:15:42,530 The end. 181 00:15:43,140 --> 00:15:46,750 Turn. To be more mathematical 182 00:15:46,750 --> 00:15:51,368 about it. And one of the things that we'd like to be able to do 183 00:15:51,368 --> 00:15:54,566 with a sequence of numbers like this is get to a series. In 184 00:15:54,566 --> 00:15:58,256 other words, to be able to add them up. So let's have a look at 185 00:15:58,256 --> 00:16:05,693 that. So SN the some of these end terms is A plus A+B plus 186 00:16:05,693 --> 00:16:12,847 A plus 2B plus. But I want just to stop there and what I 187 00:16:12,847 --> 00:16:19,490 want to do is I want to start at the end. This end 188 00:16:19,490 --> 00:16:24,089 now now the last one will be plus L. 189 00:16:25,150 --> 00:16:29,518 So what will be the next one back when we generate each term 190 00:16:29,518 --> 00:16:35,230 by adding on D. So we added on D to this one to get L. So this 191 00:16:35,230 --> 00:16:37,582 one's got to be L minus D. 192 00:16:38,920 --> 00:16:45,750 And the one before that one similarly will be L 193 00:16:45,750 --> 00:16:51,520 minus 2D. On the rest of the terms will be in between. 194 00:16:52,640 --> 00:16:54,999 Now I'm going to use a trick. 195 00:16:55,540 --> 00:16:58,850 Mathematicians often use. I'm going to write this down the 196 00:16:58,850 --> 00:17:02,072 other way around. So I have L 197 00:17:02,072 --> 00:17:05,870 there. Plus L minus 198 00:17:05,870 --> 00:17:12,433 D. Plus L minus 2D plus plus. Now what will I have? 199 00:17:12,433 --> 00:17:17,869 Well, writing this down either way around, I'll Have A at the 200 00:17:17,869 --> 00:17:21,255 end. Then I'll have this next 201 00:17:21,255 --> 00:17:23,730 term a. Plus D. 202 00:17:24,230 --> 00:17:28,973 And I'll have this next term, A plus 2D. 203 00:17:31,490 --> 00:17:36,319 Now I'm going to add these two together. Let's look what 204 00:17:36,319 --> 00:17:41,587 happens if I add SN&SN together. I've just got two of them. 205 00:17:42,970 --> 00:17:49,548 By ad A&L together I get a plus L let me just group 206 00:17:49,548 --> 00:17:50,560 those together. 207 00:17:51,820 --> 00:17:59,004 Now I've got a plus D&L Minus D, so if I add them together I have 208 00:17:59,004 --> 00:18:06,188 a plus L Plus D minus D, so all I've got left is A plus L. 209 00:18:07,290 --> 00:18:14,192 But the same thing is going to happen here. I have a plus L 210 00:18:14,192 --> 00:18:19,122 Plus 2D Takeaway 2D, so again just a plus L. 211 00:18:20,030 --> 00:18:25,070 When we get down To this end, it's still the same thing 212 00:18:25,070 --> 00:18:30,110 happening. I've A plus L takeaway 2D add onto D so again 213 00:18:30,110 --> 00:18:35,570 the DS have disappeared. If you like and I've got L plus A. 214 00:18:36,480 --> 00:18:43,884 Plus a plusle takeaway D add on DLA and right at the 215 00:18:43,884 --> 00:18:46,969 end. L plus a again. 216 00:18:48,640 --> 00:18:53,333 Well, how many of these have I got? But I've got N terms. 217 00:18:54,130 --> 00:19:01,732 In each of these lines of sums, so I must still have end terms 218 00:19:01,732 --> 00:19:07,705 here, and so this must be an times a plus L. 219 00:19:08,390 --> 00:19:15,530 And so if we now divide both sides by two, we have. 220 00:19:15,530 --> 00:19:22,670 SN is 1/2 of N times by a plus L and that 221 00:19:22,670 --> 00:19:28,620 gives us our some of the terms of an arithmetic 222 00:19:28,620 --> 00:19:35,409 progression. Let's just write down again the two results that 223 00:19:35,409 --> 00:19:42,741 we've got. We've got L the end term, or the final term 224 00:19:42,741 --> 00:19:50,073 is equal to a plus N minus one times by D and 225 00:19:50,073 --> 00:19:53,739 we've got the SN is 1/2. 226 00:19:54,320 --> 00:20:01,380 Times by N number of terms times by a plus 227 00:20:01,380 --> 00:20:02,086 L. 228 00:20:03,290 --> 00:20:09,062 Now, one thing we can do is take this expression for L and 229 00:20:09,062 --> 00:20:10,838 substitute it into here. 230 00:20:11,770 --> 00:20:19,570 Replacing this al, so let's do that. SN is equal to 1/2. 231 00:20:20,180 --> 00:20:27,716 Times by N number of terms times by a plus and instead 232 00:20:27,716 --> 00:20:35,252 of L will write this a plus N minus one times by 233 00:20:35,252 --> 00:20:39,558 D. April say gives us two way. 234 00:20:40,600 --> 00:20:47,640 So the sum of the end terms is 1/2 an 235 00:20:47,640 --> 00:20:51,160 2A plus N minus 1D. 236 00:20:51,930 --> 00:20:53,958 Close the bracket. 237 00:20:55,730 --> 00:21:02,064 And these. That I'm underlining are the three 238 00:21:02,064 --> 00:21:05,838 important things about an arithmetic progression. 239 00:21:07,280 --> 00:21:11,070 If A is the first 240 00:21:11,070 --> 00:21:16,826 term. And D is the common difference. 241 00:21:17,750 --> 00:21:23,102 And N is the 242 00:21:23,102 --> 00:21:27,116 number of terms. 243 00:21:28,370 --> 00:21:33,545 In our arithmetic progression, then, this expression gives us 244 00:21:33,545 --> 00:21:39,870 the NTH or the last term. This expression gives us the 245 00:21:39,870 --> 00:21:46,195 some of those N terms, and this expression gives us also 246 00:21:46,195 --> 00:21:49,645 the sum of the end terms. 247 00:21:50,560 --> 00:21:54,520 One of the things that you also need to understand is that 248 00:21:54,520 --> 00:21:58,150 sometimes we like to shorten the language as well as using 249 00:21:58,150 --> 00:22:02,861 algebra. So that rather than keep saying arithmetic 250 00:22:02,861 --> 00:22:07,838 progression, we often refer to these as a peas. 251 00:22:09,130 --> 00:22:11,278 Now we've got some facts, some 252 00:22:11,278 --> 00:22:16,526 information there. So let's have a look at trying to see if we 253 00:22:16,526 --> 00:22:18,662 can use them to solve some 254 00:22:18,662 --> 00:22:25,800 questions. So let's have a look at this 255 00:22:25,800 --> 00:22:31,750 sequence of numbers again, which we've identified. 256 00:22:33,460 --> 00:22:37,016 And let's ask ourselves what's the sum? 257 00:22:38,150 --> 00:22:41,100 Of. The first 258 00:22:41,780 --> 00:22:48,278 50 terms So we could start to try and add 259 00:22:48,278 --> 00:22:54,374 them up. 1 + 3 is four and four and five is 9, and nine and 260 00:22:54,374 --> 00:22:59,708 Seven is 16 and 16 and 9025, and then the next get or getting 261 00:22:59,708 --> 00:23:03,899 rather complicated. But we can write down some facts about this 262 00:23:03,899 --> 00:23:08,471 straight away. We can write down that the first term is one. 263 00:23:09,070 --> 00:23:14,350 We can write down that the common difference Dean is 2 and 264 00:23:14,350 --> 00:23:20,070 we can write down the number of terms we're dealing with. An is 265 00:23:20,070 --> 00:23:27,041 50. We know we have a formula that says SN is 1/2 266 00:23:27,041 --> 00:23:29,696 times the number of terms. 267 00:23:30,710 --> 00:23:37,696 Times 2A plus N minus 1D. So instead of having to add this up 268 00:23:37,696 --> 00:23:43,684 as though it was a big arithmetic sum a big problem, we 269 00:23:43,684 --> 00:23:48,674 can simply substitute the numbers into the formula. So SNS 270 00:23:48,674 --> 00:23:52,666 50 in this case is equal to 1/2. 271 00:23:53,080 --> 00:23:55,288 Times by 50. 272 00:23:55,800 --> 00:24:02,940 Times by two A That's just two 2 * 1 plus N minus one 273 00:24:02,940 --> 00:24:06,510 and is 50, so N minus one 274 00:24:06,510 --> 00:24:10,210 is 49. Times by the common 275 00:24:10,210 --> 00:24:11,270 difference too. 276 00:24:12,260 --> 00:24:19,716 So. We can cancel a 2 into the 50 277 00:24:19,716 --> 00:24:23,230 that gives us 25 times by now. 278 00:24:23,770 --> 00:24:30,840 2 * 49 or 2 * 49 is 98 and two is 100, so 279 00:24:30,840 --> 00:24:37,405 we have 25 times by 100, so that's 2500. So what was going 280 00:24:37,405 --> 00:24:42,455 to be quite a lengthy and difficult calculation's come out 281 00:24:42,455 --> 00:24:48,702 quite quickly. Let's see if we can solve a more difficult 282 00:24:48,702 --> 00:24:53,000 problem. 1. 283 00:24:54,110 --> 00:24:58,840 Plus 3.5. +6. 284 00:25:00,050 --> 00:25:03,620 Plus 8.5. Plus 285 00:25:04,490 --> 00:25:08,270 Plus 101. 286 00:25:10,290 --> 00:25:11,310 Add this up. 287 00:25:12,490 --> 00:25:19,552 Well. Can we identify what kind of a series this is? We can 288 00:25:19,552 --> 00:25:25,155 see quite clearly that one to 3.5 while that's a gap of 2.5 289 00:25:25,155 --> 00:25:32,051 and then a gap of 2.5 to 6. So what we've got here is in fact 290 00:25:32,051 --> 00:25:36,792 an arithmetic progression, and we can see here. We've got 100 291 00:25:36,792 --> 00:25:42,826 and one at the end. Our last term is 101 and the first term 292 00:25:42,826 --> 00:25:45,843 is one. Now we know a formula. 293 00:25:45,890 --> 00:25:49,390 For the last term L. 294 00:25:50,050 --> 00:25:57,133 Equals A plus N minus one times by D. 295 00:25:58,350 --> 00:26:05,126 Might just have a look at what we know in this formula. What we 296 00:26:05,126 --> 00:26:07,062 know L it's 101. 297 00:26:07,070 --> 00:26:12,227 We know a It's the first term, it's one. 298 00:26:13,280 --> 00:26:19,259 Plus Well, we have no idea what any is. We don't know how many 299 00:26:19,259 --> 00:26:24,061 terms we've got, so that's N minus one times by D and we know 300 00:26:24,061 --> 00:26:25,776 what that is, that's 2.5. 301 00:26:26,510 --> 00:26:31,790 Well, this is nothing more than an equation for an, so let's 302 00:26:31,790 --> 00:26:37,510 begin by taking one from each side. That gives us 100 equals N 303 00:26:37,510 --> 00:26:43,230 minus one times by 2.5. And now I'm going to divide both sides 304 00:26:43,230 --> 00:26:49,830 by 2.5 and that will give me 40 equals N minus one, and now I'll 305 00:26:49,830 --> 00:26:57,310 add 1 to both sides and so 41 is equal to end, so I know how many 306 00:26:57,310 --> 00:27:04,198 terms that. Are in this series, So what I can do now is I 307 00:27:04,198 --> 00:27:10,186 can add it up because the sum of N terms is 1/2. 308 00:27:10,860 --> 00:27:13,872 NA plus 309 00:27:13,872 --> 00:27:20,800 L. And I now know all these terms 310 00:27:20,800 --> 00:27:24,300 here have 1/2 * 41 311 00:27:24,300 --> 00:27:27,485 * 1. Plus 312 00:27:27,485 --> 00:27:34,790 101. Let me just turn the page over and write this 313 00:27:34,790 --> 00:27:36,458 some down again. 314 00:27:37,120 --> 00:27:43,564 SN is equal to 1/2 * 315 00:27:43,564 --> 00:27:48,934 41 * 1 + 101. 316 00:27:50,040 --> 00:27:57,432 So we have 1/2 times by 41 times by 102 and we can cancel 317 00:27:57,432 --> 00:28:04,824 it to there to give US 41 times by 51. And to do that 318 00:28:04,824 --> 00:28:10,632 I'd want to get out my Calculator, but we'll leave it 319 00:28:10,632 --> 00:28:12,744 there to be finished. 320 00:28:13,270 --> 00:28:16,858 So that's one kind of problem. 321 00:28:17,800 --> 00:28:21,877 Let's have a look at another kind of problem. 322 00:28:22,430 --> 00:28:28,214 Let's say we've got an arithmetic progression whose 323 00:28:28,214 --> 00:28:31,106 first term is 3. 324 00:28:32,170 --> 00:28:35,530 And the sum. 325 00:28:36,200 --> 00:28:37,570 Of. 326 00:28:39,210 --> 00:28:42,970 The first 8. 327 00:28:44,740 --> 00:28:45,630 Terms. 328 00:28:47,020 --> 00:28:53,972 Is twice. The sum 329 00:28:53,972 --> 00:28:59,736 of the first 5 330 00:28:59,736 --> 00:29:01,177 terms. 331 00:29:02,570 --> 00:29:04,820 And that seems really quite 332 00:29:04,820 --> 00:29:09,228 complicated. But it needn't be, but remember this is the 333 00:29:09,228 --> 00:29:10,374 same arithmetic progression. 334 00:29:12,250 --> 00:29:18,018 So let's have a think what this is telling us A is equal to 335 00:29:18,018 --> 00:29:23,374 three and the sum of the first 8 terms. Well, to begin with, 336 00:29:23,374 --> 00:29:28,318 let's write down what the sum of the first 8 terms is. 337 00:29:28,870 --> 00:29:31,818 Well, it's a half. 338 00:29:32,440 --> 00:29:39,088 Times N Times 2A plus and 339 00:29:39,088 --> 00:29:41,304 minus 1D. 340 00:29:42,470 --> 00:29:45,848 And N is equal to 8. 341 00:29:46,930 --> 00:29:48,750 So we've got a half. 342 00:29:49,980 --> 00:29:57,630 Times 8. 2A plus N minus one is 343 00:29:57,630 --> 00:30:05,484 7D. So S 8 is equal to half of 344 00:30:05,484 --> 00:30:11,828 eight is 4 * 2 A Plus 7D. 345 00:30:12,970 --> 00:30:20,434 But we also know that a is equal to three, so we 346 00:30:20,434 --> 00:30:27,898 can put that in there as well. That's 4 * 6 because 347 00:30:27,898 --> 00:30:31,630 a is 3 + 7 D. 348 00:30:32,340 --> 00:30:39,812 Next one, the sum of the first 5 349 00:30:39,812 --> 00:30:46,834 terms. Let me just write down some of the first 350 00:30:46,834 --> 00:30:48,862 8 terms were. 351 00:30:49,150 --> 00:30:55,554 4. Times 6 minus 352 00:30:55,554 --> 00:31:01,146 plus 7D first 5 353 00:31:01,146 --> 00:31:08,706 terms. Half times the number of terms. That's 5 * 2 A plus 354 00:31:08,706 --> 00:31:15,888 N minus one times by D will. That must be 4 because any is 355 00:31:15,888 --> 00:31:17,940 5 times by D. 356 00:31:18,900 --> 00:31:25,478 So much is 5 over 2 and let's remember that a is equal 357 00:31:25,478 --> 00:31:32,562 to three, so that 6 + 4 D. So I've got S 8 and 358 00:31:32,562 --> 00:31:39,140 I've got S5 and the question said that S8 was equal to twice 359 00:31:39,140 --> 00:31:42,970 as five. So I can write this 360 00:31:42,970 --> 00:31:48,635 for S8. Is equal to 361 00:31:48,635 --> 00:31:56,030 twice. This which is S five 2 * 5 over two 362 00:31:56,030 --> 00:32:02,306 6 + 4 D and what seemed a very difficult question as 363 00:32:02,306 --> 00:32:08,059 reduced itself to an ordinary linear equation in terms of D. 364 00:32:08,059 --> 00:32:14,335 So we can do some cancelling there and we can multiply out 365 00:32:14,335 --> 00:32:21,657 the brackets for six is a 24 + 28, D is equal to 56R. 366 00:32:21,690 --> 00:32:24,974 30 + 5 fours 367 00:32:24,974 --> 00:32:31,870 are 20D. I can take 20D from each side that gives me 368 00:32:31,870 --> 00:32:33,256 8 D there. 369 00:32:33,830 --> 00:32:40,998 And I can take 24 from each side, giving me six there. So D 370 00:32:40,998 --> 00:32:42,534 is equal to. 371 00:32:43,650 --> 00:32:49,994 Dividing both sides by 8, six over 8 or 3/4 so I know 372 00:32:49,994 --> 00:32:54,874 everything now that I could possibly want to know about 373 00:32:54,874 --> 00:32:56,338 this arithmetic progression. 374 00:32:57,940 --> 00:33:04,142 Now let's go on and have a look at our second type of special 375 00:33:04,142 --> 00:33:05,471 sequence, a geometric 376 00:33:05,471 --> 00:33:11,398 progression. So. Take these 377 00:33:11,398 --> 00:33:14,666 two six 378 00:33:14,666 --> 00:33:20,680 1854. Let's have a look at how this sequence of numbers 379 00:33:20,680 --> 00:33:23,207 is growing. We have two. Then we 380 00:33:23,207 --> 00:33:31,060 have 6. And then we have 18. Well 326 and three sixes 381 00:33:31,060 --> 00:33:38,896 are 18 and three eighteens are 54. So this sequence is growing 382 00:33:38,896 --> 00:33:45,426 by multiplying by three each time. What about this sequence 383 00:33:45,426 --> 00:33:48,436 one? Minus 384 00:33:48,436 --> 00:33:52,215 2 four. Minus 385 00:33:52,215 --> 00:33:57,040 8. What's happening here? We can see the signs are alternating, 386 00:33:57,040 --> 00:33:58,996 but let's just look at the 387 00:33:58,996 --> 00:34:05,067 numbers. 1 * 2 would be two 2 * 2 would be four. 2 * 4 would be 388 00:34:05,067 --> 00:34:10,890 8. But if we made that minus two, then one times minus two 389 00:34:10,890 --> 00:34:17,945 would be minus 2 - 2 times minus two would be plus 4 + 4 times by 390 00:34:17,945 --> 00:34:23,340 minus two would be minus 8, so this sequence to be generated is 391 00:34:23,340 --> 00:34:27,905 being multiplied by minus two. Each term is multiplied by minus 392 00:34:27,905 --> 00:34:30,395 two to give the next term. 393 00:34:31,210 --> 00:34:36,840 These are examples of geometric progressions, or if you like, 394 00:34:36,840 --> 00:34:42,830 GPS. Let's try and write one down in general using some 395 00:34:42,830 --> 00:34:48,810 algebra. So like the AP, we take A to be the first term. 396 00:34:49,640 --> 00:34:54,249 Now we need something like D. The common difference, but what 397 00:34:54,249 --> 00:35:00,115 we use is the letter R and we call it the common ratio, and 398 00:35:00,115 --> 00:35:05,143 that's the number that does the multiplying of each term to give 399 00:35:05,143 --> 00:35:06,400 the next term. 400 00:35:07,090 --> 00:35:14,395 So 3 times by two gives us 6, so that's the R. In this case 401 00:35:14,395 --> 00:35:18,291 the three. So we do a Times by 402 00:35:18,291 --> 00:35:24,999 R. And then we multiply by, in this case by three again 3 times 403 00:35:24,999 --> 00:35:29,955 by 6 gives 18, so we multiply by R again, AR squared. 404 00:35:30,670 --> 00:35:38,122 And then we multiply by three again to give us the 54. 405 00:35:38,122 --> 00:35:41,848 So by our again AR cubed. 406 00:35:42,680 --> 00:35:49,638 And what's our end term in this case? While A is the first term 407 00:35:49,638 --> 00:35:56,596 8 times by R, is the second term 8 times by R-squared is the 408 00:35:56,596 --> 00:36:03,554 third term 8 times by R cubed? Is the fourth term, so it's a 409 00:36:03,554 --> 00:36:10,015 times by R to the N minus one. Because this power there's a 410 00:36:10,015 --> 00:36:12,997 one. There is always one less. 411 00:36:13,000 --> 00:36:17,268 And the number of the term, then its position in the 412 00:36:17,268 --> 00:36:22,700 sequence. And this is the end term, so it's a Times my R to 413 00:36:22,700 --> 00:36:24,252 the N minus one. 414 00:36:25,470 --> 00:36:31,766 What about adding up a geometric progression? Let's 415 00:36:31,766 --> 00:36:39,636 write that down. SN is equal to a plus R 416 00:36:39,636 --> 00:36:41,997 Plus R-squared Plus. 417 00:36:42,580 --> 00:36:50,070 Plus AR to the N minus one, and that's the sum of N terms. 418 00:36:50,810 --> 00:36:56,342 Going to use another trick similar but not the same to what 419 00:36:56,342 --> 00:37:00,491 we did with arithmetic progressions. What I'm going to 420 00:37:00,491 --> 00:37:05,562 do is I'm going to multiply everything by the common ratio. 421 00:37:06,590 --> 00:37:11,894 So I've multiplied SN by are going to multiply this one by R, 422 00:37:11,894 --> 00:37:17,198 but I'm not going to write the answer there. I'm going to write 423 00:37:17,198 --> 00:37:23,726 it here so I've a Times by R and I've written it there plus now I 424 00:37:23,726 --> 00:37:29,030 multiply this one by R and that would give me a R-squared. I'm 425 00:37:29,030 --> 00:37:31,070 going to write it there. 426 00:37:31,630 --> 00:37:36,089 So that term is being multiplied by R and it's gone to their 427 00:37:36,089 --> 00:37:40,205 that's being multiplied by R and it's gone to their. This one 428 00:37:40,205 --> 00:37:45,350 will be multiplied by R and it will be a R cubed and it will 429 00:37:45,350 --> 00:37:46,722 have gone to their. 430 00:37:47,360 --> 00:37:52,427 Plus etc plus, and we think about what's happening. 431 00:37:53,350 --> 00:37:58,250 That term will come to here and it will look just like that one. 432 00:37:59,010 --> 00:38:03,582 Plus and then we need to multiply this by R, and that's 433 00:38:03,582 --> 00:38:07,011 another. Are that we're multiplying by, so that means 434 00:38:07,011 --> 00:38:09,297 that becomes AR to the N. 435 00:38:10,230 --> 00:38:16,548 Now look at why I've lined these up AR, AR, AR squared. Our 436 00:38:16,548 --> 00:38:19,464 squared, al, cubed, cubed and so 437 00:38:19,464 --> 00:38:25,423 on. So let's take these two lines of algebra away from each 438 00:38:25,423 --> 00:38:31,625 other, so I'll have SN minus R times by SN is equal to. Now 439 00:38:31,625 --> 00:38:38,270 have nothing here to take away from a, so the a stays as it is. 440 00:38:38,270 --> 00:38:42,700 Then I've AR takeaway are, well, that's nothing. A R-squared 441 00:38:42,700 --> 00:38:46,244 takeaway R-squared? That's nothing again, same there. And 442 00:38:46,244 --> 00:38:50,231 so on and so on. AR to the N 443 00:38:50,231 --> 00:38:55,054 minus one. Take away a art. The end minus one nothing and then 444 00:38:55,054 --> 00:38:57,532 at the end I have nothing there 445 00:38:57,532 --> 00:39:02,018 take away. AR to the N. 446 00:39:03,240 --> 00:39:07,049 Now I need to look closely at both sides of what I've got 447 00:39:07,049 --> 00:39:10,858 written down, and I'm going to turn this over and write it down 448 00:39:10,858 --> 00:39:18,516 again. So we've SN minus RSN is equal to A. 449 00:39:19,020 --> 00:39:22,240 Minus AR to the N. 450 00:39:22,930 --> 00:39:28,429 Now here I've got a common factor SN the some of the end 451 00:39:28,429 --> 00:39:34,351 terms when I take that out, I've won their minus R of them there, 452 00:39:34,351 --> 00:39:41,542 so I get SN times by one minus R is equal 2 and here I've got a 453 00:39:41,542 --> 00:39:48,310 common Factor A and I can take a out giving me one minus R to the 454 00:39:48,310 --> 00:39:53,809 N. Remember it was the sum of N terms that I wanted so. 455 00:39:53,860 --> 00:40:00,466 SN is equal to a Times 1 minus R to the N and to get the SN on 456 00:40:00,466 --> 00:40:05,604 its own, I've had to divide by one minus R, so I must divide 457 00:40:05,604 --> 00:40:07,439 this by one minus R. 458 00:40:09,370 --> 00:40:15,844 And that's my formula for the sum of N terms of a geometric 459 00:40:15,844 --> 00:40:19,828 progression. And let's just remind ourselves what the 460 00:40:19,828 --> 00:40:24,808 symbols are N is equal to the number of terms. 461 00:40:24,820 --> 00:40:30,847 A is the first term of our 462 00:40:30,847 --> 00:40:34,291 geometric progression and are 463 00:40:34,291 --> 00:40:40,318 we said was called the common ratio. 464 00:40:41,420 --> 00:40:48,011 OK, and let's just remember the NTH term in the sequence was AR 465 00:40:48,011 --> 00:40:55,109 to the N minus one. So those are our fax so far about GPS 466 00:40:55,109 --> 00:41:00,686 or geometric progressions. Let's see if we can use these facts 467 00:41:00,686 --> 00:41:07,277 in order to be able to help us solve some problems and do 468 00:41:07,277 --> 00:41:15,027 some questions. So first of all, let's take this 2 + 6 + 469 00:41:15,027 --> 00:41:22,268 18 + 54 plus. Let's say there are six terms. What's the answer 470 00:41:22,268 --> 00:41:25,610 when it comes to adding those 471 00:41:25,610 --> 00:41:32,550 up? Well, we know that a is equal to two. We know that 472 00:41:32,550 --> 00:41:39,825 our is equal to three and we know that N is equal to six. So 473 00:41:39,825 --> 00:41:47,585 to solve that, all we need to do is write down that the sum of N 474 00:41:47,585 --> 00:41:54,860 terms is a Times 1 minus R to the N all over 1 minus R. 475 00:41:54,860 --> 00:41:57,285 Substitute our numbers in two 476 00:41:57,285 --> 00:42:03,904 times. 1 - 3 to the power 6. 477 00:42:04,520 --> 00:42:07,991 Over 1 - 478 00:42:07,991 --> 00:42:14,612 3. So this is 2 * 1 - 3 to the power six over minus 479 00:42:14,612 --> 00:42:20,044 two, and we can cancel a minus two with the two that we leave 480 00:42:20,044 --> 00:42:24,700 as with a minus one there and one there if I multiply 481 00:42:24,700 --> 00:42:29,744 throughout by the minus one, I'll have minus 1 * 1 is minus 482 00:42:29,744 --> 00:42:35,952 one and minus one times minus 3 to the six is 3 to the 6th, so 483 00:42:35,952 --> 00:42:39,444 the sum of N terms is 3 to the 484 00:42:39,444 --> 00:42:45,430 power 6. Minus one and with a Calculator we could workout what 485 00:42:45,430 --> 00:42:49,056 3 to the power 6 - 1 486 00:42:49,056 --> 00:42:52,256 was. Let's take 487 00:42:52,256 --> 00:42:59,440 another. Question to do with summing the terms of a geometric 488 00:42:59,440 --> 00:43:05,892 progression. What's the sum of that? Let's say for five 489 00:43:05,892 --> 00:43:11,772 terms. While we can begin by identifying the first term, 490 00:43:11,772 --> 00:43:15,888 that's eight, and what's the common ratio? 491 00:43:17,100 --> 00:43:23,100 Well, to go from 8 to 4 as a number we would have it, but 492 00:43:23,100 --> 00:43:27,900 there's a minus sign in there. So that suggests that the common 493 00:43:27,900 --> 00:43:32,700 ratio is minus 1/2. Let's just check it minus four times. By 494 00:43:32,700 --> 00:43:39,100 minus 1/2 is plus 2 + 2 times Y minus 1/2 is minus one, and we 495 00:43:39,100 --> 00:43:41,900 said five terms, so Ann is equal 496 00:43:41,900 --> 00:43:49,494 to 5. So we can write down our formula. SN is equal to 497 00:43:49,494 --> 00:43:57,390 a Times 1 minus R to the power N all over 1 minus R. 498 00:43:57,980 --> 00:44:01,120 And so A is 8. 499 00:44:02,560 --> 00:44:10,220 1 minus and this is minus 1/2 to the power 500 00:44:10,220 --> 00:44:16,460 5. All over 1 minus minus 1/2. You can see these 501 00:44:16,460 --> 00:44:19,328 questions get quite complicated with the 502 00:44:19,328 --> 00:44:24,108 arithmetic, so you have to be very careful and you 503 00:44:24,108 --> 00:44:27,932 have to have a good knowledge of fractions. 504 00:44:29,320 --> 00:44:37,112 This is 8 * 1. Now let's have a look at minus 1/2 to the power 505 00:44:37,112 --> 00:44:42,360 5. Well, I'm multiplying the minus sign by itself five times, 506 00:44:42,360 --> 00:44:47,136 which would give me a negative number, and I've got a minus 507 00:44:47,136 --> 00:44:51,912 sign there outside the bracket. That's going to mean I've got 6 508 00:44:51,912 --> 00:44:57,086 minus signs together. Makes it plus. So now I can look at the 509 00:44:57,086 --> 00:44:59,076 half to the power 5. 510 00:44:59,670 --> 00:45:02,574 Well, that's going to be one 511 00:45:02,574 --> 00:45:09,270 over. 248-1632 to to the power 512 00:45:09,270 --> 00:45:12,696 five is 32. 513 00:45:13,400 --> 00:45:20,504 All over 1 minus minus 1/2. That's 1 + 1/2. Let's write 514 00:45:20,504 --> 00:45:23,464 that as three over 2. 515 00:45:24,540 --> 00:45:27,170 So this is equal to. 516 00:45:28,160 --> 00:45:30,700 Now I've got 8. 517 00:45:31,450 --> 00:45:37,764 Times by one plus, one over 32, and I'm dividing by three over 2 518 00:45:37,764 --> 00:45:42,725 to divide by a fraction. We invert the fraction that's two 519 00:45:42,725 --> 00:45:49,039 over 3 and we multiply by and we just turn the page to finish 520 00:45:49,039 --> 00:45:50,392 this one off. 521 00:45:51,110 --> 00:45:58,840 So we have SN is equal to 8 * 1 522 00:45:58,840 --> 00:46:06,570 + 1 over 32 times by 2/3 is equal to 523 00:46:06,570 --> 00:46:14,300 8 times by now one and 132nd. Well, there are 524 00:46:14,300 --> 00:46:22,030 3230 seconds in one, so altogether there I've got 33. 525 00:46:22,030 --> 00:46:25,264 30 seconds times 526 00:46:25,264 --> 00:46:30,926 by 2/3. And we can do some canceling threes 527 00:46:30,926 --> 00:46:35,582 into 30. Three will go 11 and threes into three. There goes 528 00:46:35,582 --> 00:46:42,536 one. Twos into two goes one and tools into 32, goes 16 and 18 529 00:46:42,536 --> 00:46:48,864 two eight goes one and eight into 16 goes 2. So we 1 * 530 00:46:48,864 --> 00:46:55,192 11 * 1 that's just 11 over 2 because we've 2 times by one 531 00:46:55,192 --> 00:47:01,068 there. So we love Nova two or we prefer five and a half. 532 00:47:01,840 --> 00:47:07,216 So that we've got the some of those five terms of that 533 00:47:07,216 --> 00:47:12,144 particular GP. Five and a half, 11 over 2 or 5.5. 534 00:47:12,680 --> 00:47:19,775 But here's a different question. What if we've got the sequence 535 00:47:19,775 --> 00:47:27,512 248? 128 how many terms are we got? How many bits do we 536 00:47:27,512 --> 00:47:33,944 need to get from 2 up to 128? Well, let's begin by 537 00:47:33,944 --> 00:47:36,088 identifying the first term 538 00:47:36,088 --> 00:47:39,610 that's two. This is. 539 00:47:40,820 --> 00:47:45,792 A geometric progression because we multiply by two to get each 540 00:47:45,792 --> 00:47:52,120 term. So the common ratio are is 2 and what we don't know is 541 00:47:52,120 --> 00:47:59,168 what's N. So let's have a look. This is the last term and we 542 00:47:59,168 --> 00:48:05,524 know our expression for the last term. 128 is equal to AR to the 543 00:48:05,524 --> 00:48:06,886 N minus one. 544 00:48:07,700 --> 00:48:14,427 So let's substituting some of our information. 545 00:48:14,450 --> 00:48:22,310 A is 2 times by two 4R to the N minus one. 546 00:48:23,010 --> 00:48:29,020 Well, we can divide both sides by this two here, 547 00:48:29,020 --> 00:48:31,424 which will give us. 548 00:48:32,570 --> 00:48:38,240 64 is equal to two to the N minus one. 549 00:48:38,870 --> 00:48:45,170 I think about that it's 248 sixteen 3264 so I had to 550 00:48:45,170 --> 00:48:51,995 multiply 2 by itself six times in order to get 64, so 2 551 00:48:51,995 --> 00:48:59,345 to the power 6, which is 64 is equal to 2 to the power 552 00:48:59,345 --> 00:49:06,695 N minus one, so six is equal to N minus one, and so N 553 00:49:06,695 --> 00:49:09,845 is equal to 7, adding one. 554 00:49:09,860 --> 00:49:14,960 To each side. In other words, there were Seven terms in our. 555 00:49:15,520 --> 00:49:22,122 Geometric progression. Type of question that's often given 556 00:49:22,122 --> 00:49:26,698 for geometric progressions is given a geometric progression. 557 00:49:26,698 --> 00:49:32,990 How many terms do you need to add together before you 558 00:49:32,990 --> 00:49:38,710 exceed a certain limit? So, for instance, here's a geometric. 559 00:49:38,820 --> 00:49:44,832 Progression. How many times of this geometric progression do we 560 00:49:44,832 --> 00:49:50,684 need to act together in order to be sure that the some of them 561 00:49:50,684 --> 00:49:52,356 will get over 20? 562 00:49:53,040 --> 00:49:58,932 Well, first of all, let's try and identify this as a geometric 563 00:49:58,932 --> 00:50:04,333 progression. The first term is on and it looks like what's 564 00:50:04,333 --> 00:50:09,243 doing the multiplying. The common ratio is 1.1. Let's just 565 00:50:09,243 --> 00:50:10,716 check that here. 566 00:50:11,290 --> 00:50:16,919 1.1 times by one point, one well. That's kind of like 11 * 567 00:50:16,919 --> 00:50:18,218 11 is 121. 568 00:50:18,730 --> 00:50:23,086 With two numbers after the decimal point in one point 1 * 569 00:50:23,086 --> 00:50:27,442 1.1 and with two numbers after the decimal point there. So yes, 570 00:50:27,442 --> 00:50:29,257 this is a geometric progression. 571 00:50:30,050 --> 00:50:36,719 So let's write down our formula for N terms sum of N terms 572 00:50:36,719 --> 00:50:42,362 is equal to a Times 1 minus R to the N. 573 00:50:42,970 --> 00:50:50,320 All over 1 minus R. We want to know what value of N is just 574 00:50:50,320 --> 00:50:53,260 going to take us over 20. 575 00:50:53,880 --> 00:51:00,090 So let's substituting some numbers. This is one for 576 00:51:00,090 --> 00:51:03,540 a 1 - 1.1 to 577 00:51:03,540 --> 00:51:10,816 the N. All over 1 - 1.1 that 578 00:51:10,816 --> 00:51:15,952 has to be greater than 20. 579 00:51:17,210 --> 00:51:22,886 So one times by that isn't going to affect what's in the 580 00:51:22,886 --> 00:51:29,981 brackets. That would be 1 - 1.1 to the N all over 1 - 1.1 581 00:51:29,981 --> 00:51:35,184 is minus nought. .1 that has to be greater than 20. 582 00:51:36,510 --> 00:51:42,256 Now if I use the minus sign wisely. In other words, If I 583 00:51:42,256 --> 00:51:44,024 divide if you like. 584 00:51:45,280 --> 00:51:48,724 Minus note .1 into there as 585 00:51:48,724 --> 00:51:52,110 a. Division, then I'll have. 586 00:51:52,690 --> 00:51:58,878 The minus sign will make that a minus and make that a plus, so 587 00:51:58,878 --> 00:52:05,066 I'll have one point 1 to the N minus one and divided by North 588 00:52:05,066 --> 00:52:10,370 Point one is exactly the same as multiplying by 10. That means 589 00:52:10,370 --> 00:52:12,580 I've got a 10 here. 590 00:52:13,310 --> 00:52:16,467 That I can divide both sides by. 591 00:52:17,620 --> 00:52:23,613 So let's just write this down again 1.1 to the N minus one 592 00:52:23,613 --> 00:52:29,606 times by 10 has to be greater than 20. So let's divide both 593 00:52:29,606 --> 00:52:36,521 sides by 10, one point 1 to the N minus one has to be greater 594 00:52:36,521 --> 00:52:43,897 than two and will add the one to both sides 1.1 to the end has to 595 00:52:43,897 --> 00:52:45,741 be greater than three. 596 00:52:46,550 --> 00:52:52,010 Problem how do we find N? One of the ways of solving equations 597 00:52:52,010 --> 00:52:57,470 like this is to take logarithms of both sides, so I'm going to 598 00:52:57,470 --> 00:53:02,510 take natural logarithms of both sides. I'm going to do it to 599 00:53:02,510 --> 00:53:07,130 this site first. That's the natural logarithm of 3 N about 600 00:53:07,130 --> 00:53:12,590 this side. When you're taking a log of a number that's raised to 601 00:53:12,590 --> 00:53:17,210 the power, that's the equivalent of multiplying the log of that 602 00:53:17,210 --> 00:53:23,191 number. By the power that's N times the log of 1.1. Well 603 00:53:23,191 --> 00:53:29,704 now this is just an equation for N because N has got to 604 00:53:29,704 --> 00:53:36,217 be greater than the log of 3 divided by the log of 1.1 605 00:53:36,217 --> 00:53:37,720 because after all. 606 00:53:39,200 --> 00:53:44,645 Log of three is just a number and log of 1.1 is just a number 607 00:53:44,645 --> 00:53:48,275 and this is the sort of calculation that really does 608 00:53:48,275 --> 00:53:50,816 have to be done on a Calculator. 609 00:53:51,800 --> 00:53:57,146 So if we take our Calculator and we turn it on. 610 00:53:58,360 --> 00:54:02,370 And we do the calculation. The natural log of three. 611 00:54:03,930 --> 00:54:09,678 Divided by the natural log of 1.1, we ask our Calculator to 612 00:54:09,678 --> 00:54:16,384 calculate that for us. It tells us that it's 11.5 to 6 and some 613 00:54:16,384 --> 00:54:20,695 more decimal places. We're not really worried about these 614 00:54:20,695 --> 00:54:26,922 decimal places. An is a whole number and it has to be greater 615 00:54:26,922 --> 00:54:33,628 than 11 and some bits, so N has got to be 12 or more. 616 00:54:35,970 --> 00:54:43,458 That's one last twist to our geometric progression. 617 00:54:43,460 --> 00:54:46,900 Let's have a look at 618 00:54:46,900 --> 00:54:53,825 this one. What have we got got 619 00:54:53,825 --> 00:55:00,479 a geometric progression. First term a 620 00:55:00,479 --> 00:55:02,697 is one. 621 00:55:04,360 --> 00:55:10,910 Common ratio is 1/2 because we're multiplying by 1/2 each 622 00:55:10,910 --> 00:55:14,800 time. That write down 623 00:55:14,800 --> 00:55:21,456 some sums. S1, the sum of the first term is just. 624 00:55:22,060 --> 00:55:25,345 1. What's 625 00:55:25,345 --> 00:55:32,665 S2? That's the sum of the first 2 terms, so 626 00:55:32,665 --> 00:55:37,070 that's. Three over 2. 627 00:55:38,460 --> 00:55:43,420 What's the sum of the first three terms? That's one. 628 00:55:44,560 --> 00:55:48,106 Plus 1/2 + 629 00:55:48,106 --> 00:55:55,781 1/4. Add those up in terms of how many quarters 630 00:55:55,781 --> 00:55:59,555 have we got then that is 631 00:55:59,555 --> 00:56:06,695 7. Quarters As for the sum of 632 00:56:06,695 --> 00:56:08,505 the first. 633 00:56:08,760 --> 00:56:15,339 4. Terms. Add those up in terms of how 634 00:56:15,339 --> 00:56:21,037 many eighths if we got so we've got eight of them there. Four of 635 00:56:21,037 --> 00:56:26,328 them there. That's 12. Two of them there. That's 14 and one of 636 00:56:26,328 --> 00:56:28,363 them there. That's 15 eighths. 637 00:56:29,090 --> 00:56:31,978 Seems to be some sort of pattern here. 638 00:56:32,990 --> 00:56:36,374 Here we seem to be 1/2 short of 639 00:56:36,374 --> 00:56:42,690 two. Here we seem to 640 00:56:42,690 --> 00:56:49,410 be 1/4. Short of two here, we seem to be an eighth 641 00:56:49,410 --> 00:56:55,052 short of two and we look at the first one. Then we're clearly 1 642 00:56:55,052 --> 00:56:56,261 short of two. 643 00:56:56,930 --> 00:57:04,014 He's a powers of two. Let's have a look 2 - 2 to the 644 00:57:04,014 --> 00:57:09,580 power zero, 'cause 2 to the power zero is 1 two. 645 00:57:10,310 --> 00:57:11,510 Minus. 646 00:57:12,750 --> 00:57:20,660 2 to the power minus one 2 - 2 to the power minus two 647 00:57:20,660 --> 00:57:24,615 2 - 2 to the power minus 648 00:57:24,615 --> 00:57:29,414 three. But each of these is getting smaller. We're getting 649 00:57:29,414 --> 00:57:34,538 nearer and nearer to two. The next one we take away will be a 650 00:57:34,538 --> 00:57:40,028 16th, the one after that will be a 32nd and the next bit we take 651 00:57:40,028 --> 00:57:45,884 off 2 is going to be a 64th and then a 128 and then at one 652 00:57:45,884 --> 00:57:50,276 256th. So we're getting the bits were taking away from two are 653 00:57:50,276 --> 00:57:53,570 getting smaller and smaller and smaller until eventually we 654 00:57:53,570 --> 00:57:56,498 wouldn't be able to distinguish them from zero. 655 00:57:56,510 --> 00:58:01,592 And so if we could Add all of these up forever, a sum to 656 00:58:01,592 --> 00:58:06,674 Infinity, if you like the answer, or to be 2 or as near as 657 00:58:06,674 --> 00:58:12,482 we want to be to two. So let's see if we can have a look at 658 00:58:12,482 --> 00:58:13,934 that with some algebra. 659 00:58:14,760 --> 00:58:22,208 We know that the sum to end terms is equal to a Times 1 660 00:58:22,208 --> 00:58:27,528 minus R to the N all over 1 minus R. 661 00:58:28,050 --> 00:58:33,118 What we want to have a look at is this thing are because what 662 00:58:33,118 --> 00:58:34,566 was crucial about this? 663 00:58:35,960 --> 00:58:42,912 Geometric progression was at the common ratio was a half a 664 00:58:42,912 --> 00:58:45,440 number less than one. 665 00:58:45,450 --> 00:58:48,192 So let's have a look what 666 00:58:48,192 --> 00:58:55,928 happens. When all is bigger than one to R to the power N. 667 00:58:56,630 --> 00:59:01,778 We are is bigger than one and we keep multiplying it by itself. 668 00:59:02,280 --> 00:59:08,737 Grows, it grows very rapidly and really gets very big very 669 00:59:08,737 --> 00:59:15,781 quickly. Check it with two, 2, four, 816. It goes off til 670 00:59:15,781 --> 00:59:20,880 Infinity. And because it goes off to Infinity, it takes the 671 00:59:20,880 --> 00:59:22,875 sum with it as well. 672 00:59:24,350 --> 00:59:28,232 What about if our is equal 673 00:59:28,232 --> 00:59:34,380 to 1? Well, we can't really use this formula then because we 674 00:59:34,380 --> 00:59:39,024 would be dividing by zero. But if you think about it, are 675 00:59:39,024 --> 00:59:44,442 equals 1 means every term is the same. So if we start off with 676 00:59:44,442 --> 00:59:49,860 one every term is the same 1111 and you just add them all up. 677 00:59:49,860 --> 00:59:55,278 But again that means the sum is going to go off to Infinity if 678 00:59:55,278 --> 00:59:59,535 you take the number any number and add it to itself. 679 01:00:00,340 --> 01:00:04,504 An infinite number of times you're going to get a very, very 680 01:00:04,504 --> 01:00:12,083 big number. What happens if our is less than minus one? 681 01:00:12,083 --> 01:00:14,711 Something like minus 2? 682 01:00:15,450 --> 01:00:19,751 Well, what's going to happen then to R to the N? 683 01:00:20,920 --> 01:00:26,058 Well, it's going to be plus an. It's going to be minus as we 684 01:00:26,058 --> 01:00:31,196 multiply by this number such as minus two. So we have minus 2 + 685 01:00:31,196 --> 01:00:35,600 4 minus A. The thing to notice is it's getting bigger, it's 686 01:00:35,600 --> 01:00:40,371 getting bigger each time. So again are to the end is going to 687 01:00:40,371 --> 01:00:44,041 go off to Infinity. It's going to oscillate between plus 688 01:00:44,041 --> 01:00:48,445 Infinity and minus Infinity, but it's going to get very big and 689 01:00:48,445 --> 01:00:51,748 that means this sum is also going to get. 690 01:00:51,750 --> 01:00:52,490 Very big. 691 01:00:53,570 --> 01:01:00,086 What about our equals minus one? Well, if R equals minus one, 692 01:01:00,086 --> 01:01:05,516 let's think about a sequence like that. Well, a typical 693 01:01:05,516 --> 01:01:08,774 sequence might be 1 - 1. 694 01:01:09,490 --> 01:01:14,092 1 - 1 and we can see the problem. It depends where we 695 01:01:14,092 --> 01:01:19,402 stop. If I stop here the sum is 0 but if I put another one 696 01:01:19,402 --> 01:01:24,004 there, the sum is one. So we've got an infinite number of terms 697 01:01:24,004 --> 01:01:28,606 then. Well, it depends on money I've got us to what the answer 698 01:01:28,606 --> 01:01:33,562 is so there isn't a limit for SN. There isn't a thing that it 699 01:01:33,562 --> 01:01:35,686 can come to a definite number. 700 01:01:36,670 --> 01:01:40,963 Let's have a look. We've considered all possible values 701 01:01:40,963 --> 01:01:46,687 of our except those where are is between plus and minus one. 702 01:01:47,230 --> 01:01:50,270 Let's take our equals 1/2 as an example. 703 01:01:51,590 --> 01:01:54,446 Or half trans by half is 1/4. 704 01:01:55,470 --> 01:01:59,208 Reply by 1/2. Again that's an eighth. 705 01:02:00,410 --> 01:02:03,170 Multiply by 1/2 again, that's a 706 01:02:03,170 --> 01:02:08,846 16. Multiplied by 1/2 again, that's a 32nd. 707 01:02:09,640 --> 01:02:14,810 By half again that's a 64th by 1/2 again, that's 128. 708 01:02:16,080 --> 01:02:21,816 It's getting smaller, and if we do it enough times then it's 709 01:02:21,816 --> 01:02:24,684 going to head off till 0. 710 01:02:25,690 --> 01:02:30,530 What about a negative one? You might say, let's think about 711 01:02:30,530 --> 01:02:36,176 minus 1/2. Now multiplied by minus 1/2, it's a quarter. 712 01:02:36,176 --> 01:02:41,496 Multiply the quarter by minus 1/2. It's minus an eighth. 713 01:02:41,496 --> 01:02:46,816 Multiply again by minus 1/2. Well, that's plus a 16th. 714 01:02:46,816 --> 01:02:52,668 Multiply again by minus 1/2. That's minus a 32nd, so we're 715 01:02:52,668 --> 01:02:59,052 approaching 0, but where dotting about either side of 0 plus them 716 01:02:59,052 --> 01:03:03,840 were minus, then were plus then where mine is. 717 01:03:03,870 --> 01:03:08,562 We're getting nearer to zero each time, so again are to the 718 01:03:08,562 --> 01:03:13,645 power. N is going off to zero. What does that mean? It means 719 01:03:13,645 --> 01:03:19,119 that this some. Here we can have what we call a sum to Infinity. 720 01:03:19,119 --> 01:03:23,811 Sometimes it's just written with an S and sometimes it's got a 721 01:03:23,811 --> 01:03:25,766 little Infinity sign on it. 722 01:03:26,480 --> 01:03:32,822 What that tells us? Because this art of the end is going off to 723 01:03:32,822 --> 01:03:40,070 0 then it's a times by one over 1 minus R and that's our sum to 724 01:03:40,070 --> 01:03:42,788 Infinity. In other words, we can 725 01:03:42,788 --> 01:03:48,934 add up. An infinite number of terms for a geometric 726 01:03:48,934 --> 01:03:55,772 progression provided. The common ratio is between one and minus 727 01:03:55,772 --> 01:04:02,801 one, so let's have a look at an example. Supposing we've 728 01:04:02,801 --> 01:04:04,718 got this row. 729 01:04:04,750 --> 01:04:12,120 Metric progression. Well, first term is one 730 01:04:12,120 --> 01:04:16,524 now a common ratio is 1/3. 731 01:04:17,710 --> 01:04:19,796 And what does this come to when 732 01:04:19,796 --> 01:04:26,460 we add up? As many terms as we can, what's the sum to 733 01:04:26,460 --> 01:04:32,652 Infinity? We know the formula that's a over 1 minus R, so 734 01:04:32,652 --> 01:04:39,360 let's put the numbers in this one for a over 1 - 1/3. 735 01:04:40,460 --> 01:04:46,323 So the one on tops OK and the one minus third. Well that's 736 01:04:46,323 --> 01:04:52,186 2/3, and if we're dividing by a fraction then we invert it and 737 01:04:52,186 --> 01:04:57,147 multiply. So altogether that would come to three over 2, so 738 01:04:57,147 --> 01:04:59,853 it's very easy formula to use. 739 01:05:01,510 --> 01:05:04,246 Finally, just let's recap for a 740 01:05:04,246 --> 01:05:06,710 geometric progression. A. 741 01:05:07,360 --> 01:05:10,760 Is the first term. 742 01:05:10,760 --> 01:05:13,480 Aw. 743 01:05:14,740 --> 01:05:16,309 Is the common. 744 01:05:17,140 --> 01:05:24,026 Ratio. So a geometric progression looks like 745 01:05:24,026 --> 01:05:31,436 AARA, R-squared, AR, cubed and the N Terminus series AR 746 01:05:31,436 --> 01:05:35,141 to the N minus one. 747 01:05:35,780 --> 01:05:41,060 And if we want to add up this sequence of numbers SN. 748 01:05:41,820 --> 01:05:49,628 Then that's a Times 1 minus R to the power N or over 1 minus R. 749 01:05:50,490 --> 01:05:55,352 And if we're lucky enough to have our between plus and 750 01:05:55,352 --> 01:05:59,772 minus one, sometimes that's written as the modulus of art 751 01:05:59,772 --> 01:06:04,634 is less than one. If we're lucky to have this condition, 752 01:06:04,634 --> 01:06:10,380 then we can get a sum to Infinity, which is a over 1 753 01:06:10,380 --> 01:06:11,264 minus R.