[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.84,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.84,0:00:02.43,Default,,0000,0000,0000,,We're on problem 67.
Dialogue: 0,0:00:02.43,0:00:05.69,Default,,0000,0000,0000,,They ask us how many terms of\Nthe binomial expansion of x
Dialogue: 0,0:00:05.69,0:00:08.00,Default,,0000,0000,0000,,squared plus 2y to\Nthe third to the
Dialogue: 0,0:00:08.00,0:00:09.50,Default,,0000,0000,0000,,twentieth power contain?
Dialogue: 0,0:00:09.50,0:00:11.89,Default,,0000,0000,0000,,And so right when you see, oh\Nmy god, to the twentieth
Dialogue: 0,0:00:11.89,0:00:14.76,Default,,0000,0000,0000,,power, that will take me forever\Nto do, and then I'll
Dialogue: 0,0:00:14.76,0:00:15.93,Default,,0000,0000,0000,,have to count the terms. But\Nremember, they're just asking
Dialogue: 0,0:00:15.93,0:00:17.65,Default,,0000,0000,0000,,you how many terms.\NThey're not asking
Dialogue: 0,0:00:17.65,0:00:18.89,Default,,0000,0000,0000,,you to find the expansion.
Dialogue: 0,0:00:18.89,0:00:20.37,Default,,0000,0000,0000,,So you just have to say, oh boy,\Nlet me just think about
Dialogue: 0,0:00:20.37,0:00:21.39,Default,,0000,0000,0000,,this a little bit.
Dialogue: 0,0:00:21.39,0:00:29.56,Default,,0000,0000,0000,,And think of it this way: x plus\Ny to the first power has
Dialogue: 0,0:00:29.56,0:00:31.24,Default,,0000,0000,0000,,how many terms?
Dialogue: 0,0:00:31.24,0:00:36.03,Default,,0000,0000,0000,,It has-- well it's\Njust x plus y.
Dialogue: 0,0:00:36.03,0:00:41.91,Default,,0000,0000,0000,,And so it has 2 terms.\Nx plus y, squared.
Dialogue: 0,0:00:41.91,0:00:47.51,Default,,0000,0000,0000,,That is equal to x squared\Nplus 2xy plus y squared.
Dialogue: 0,0:00:47.51,0:00:52.91,Default,,0000,0000,0000,,So that has 3 terms. If I were\Nto do x plus e The third
Dialogue: 0,0:00:52.91,0:00:55.85,Default,,0000,0000,0000,,power, I could do a binomial\Nexpansion, I could do it with
Dialogue: 0,0:00:55.85,0:00:56.90,Default,,0000,0000,0000,,Pascal's Triangle.
Dialogue: 0,0:00:56.90,0:00:57.97,Default,,0000,0000,0000,,Let me do it with Pascal's\NTriangle.
Dialogue: 0,0:00:57.97,0:00:58.98,Default,,0000,0000,0000,,I have the 1 and the 1.
Dialogue: 0,0:00:58.98,0:01:01.24,Default,,0000,0000,0000,,That's to the first power.
Dialogue: 0,0:01:01.24,0:01:04.71,Default,,0000,0000,0000,,The second power, those\Nare the coefficients.
Dialogue: 0,0:01:04.71,0:01:08.03,Default,,0000,0000,0000,,The third power, the\Ncoefficients become 1, 3--
Dialogue: 0,0:01:08.03,0:01:12.22,Default,,0000,0000,0000,,because 1 plus 2 is 3-- 3, 1.
Dialogue: 0,0:01:12.22,0:01:14.07,Default,,0000,0000,0000,,And actually I don't even\Nhave to expand it out.
Dialogue: 0,0:01:14.07,0:01:15.59,Default,,0000,0000,0000,,Ill do it just for a\Nlittle practice.
Dialogue: 0,0:01:15.59,0:01:24.88,Default,,0000,0000,0000,,It x to the third plus 3x\Nsquared y plus 3xy squared
Dialogue: 0,0:01:24.88,0:01:25.92,Default,,0000,0000,0000,,plus y to the third.
Dialogue: 0,0:01:25.92,0:01:27.04,Default,,0000,0000,0000,,I just used these.
Dialogue: 0,0:01:27.04,0:01:31.75,Default,,0000,0000,0000,,But when you take any binomial\Nand you take it to the third
Dialogue: 0,0:01:31.75,0:01:36.10,Default,,0000,0000,0000,,power, you end up with 4 terms.\NAnd depending on how
Dialogue: 0,0:01:36.10,0:01:38.52,Default,,0000,0000,0000,,you think of it, if you think of\Nit with Pascal's Triangle,
Dialogue: 0,0:01:38.52,0:01:40.45,Default,,0000,0000,0000,,every time we go down we're\Njust going to be adding
Dialogue: 0,0:01:40.45,0:01:41.15,Default,,0000,0000,0000,,another term.
Dialogue: 0,0:01:41.15,0:01:43.91,Default,,0000,0000,0000,,When we take the fourth power\Nit's going to have 5 terms. Or
Dialogue: 0,0:01:43.91,0:01:46.45,Default,,0000,0000,0000,,when you think about, when we\Nuse the binomial expansion,
Dialogue: 0,0:01:46.45,0:01:48.62,Default,,0000,0000,0000,,and you might want to watch\Nsome videos on that.
Dialogue: 0,0:01:48.62,0:01:51.09,Default,,0000,0000,0000,,There you also have, if you're\Ntaking it to the nth power,
Dialogue: 0,0:01:51.09,0:01:53.40,Default,,0000,0000,0000,,you end up with n\Nplus 1 terms.
Dialogue: 0,0:01:53.40,0:01:57.38,Default,,0000,0000,0000,,So in this example, we have a\Nbinomial-- a binomial just
Dialogue: 0,0:01:57.38,0:01:59.13,Default,,0000,0000,0000,,means a two-term polynomial.
Dialogue: 0,0:01:59.13,0:02:01.09,Default,,0000,0000,0000,,And we're taking it to\Nthe twentieth power.
Dialogue: 0,0:02:01.09,0:02:02.90,Default,,0000,0000,0000,,If we were taking it to third\Npower, we'd have four terms.
Dialogue: 0,0:02:02.90,0:02:04.32,Default,,0000,0000,0000,,If we were taking it to the\Nfourth power, we'd have 5
Dialogue: 0,0:02:04.32,0:02:06.57,Default,,0000,0000,0000,,terms. So if we're taking it to\Nthe twentieth power, we're
Dialogue: 0,0:02:06.57,0:02:11.39,Default,,0000,0000,0000,,going to have 21 terms.\NAnd that's choice B.
Dialogue: 0,0:02:11.39,0:02:12.64,Default,,0000,0000,0000,,Next problem.
Dialogue: 0,0:02:12.64,0:02:18.42,Default,,0000,0000,0000,,
Dialogue: 0,0:02:18.42,0:02:21.89,Default,,0000,0000,0000,,All right, they want us to keep\Ndoing these and we'll do
Dialogue: 0,0:02:21.89,0:02:23.20,Default,,0000,0000,0000,,it because they want us to.
Dialogue: 0,0:02:23.20,0:02:32.40,Default,,0000,0000,0000,,
Dialogue: 0,0:02:32.40,0:02:35.54,Default,,0000,0000,0000,,All right, what are the first\Nfour terms of the expansion of
Dialogue: 0,0:02:35.54,0:02:37.54,Default,,0000,0000,0000,,1 plus 2x to the sixth power?
Dialogue: 0,0:02:37.54,0:02:39.75,Default,,0000,0000,0000,,And here we'll just use the\Nbinomial expansion because I
Dialogue: 0,0:02:39.75,0:02:41.35,Default,,0000,0000,0000,,think that's what they\Nwant us to do.
Dialogue: 0,0:02:41.35,0:02:43.35,Default,,0000,0000,0000,,And that's probably the\Nfastest way to do it.
Dialogue: 0,0:02:43.35,0:02:46.21,Default,,0000,0000,0000,,So let's just think of the\Ncoefficients first. So this is
Dialogue: 0,0:02:46.21,0:02:47.58,Default,,0000,0000,0000,,going to be-- well\Nlet's just do it.
Dialogue: 0,0:02:47.58,0:02:51.92,Default,,0000,0000,0000,,This is going to be equal\Nto 6 choose 0.
Dialogue: 0,0:02:51.92,0:02:56.07,Default,,0000,0000,0000,,We're just going to do the first\Nfour terms. Times 1 to
Dialogue: 0,0:02:56.07,0:03:00.32,Default,,0000,0000,0000,,the 6 times 2x to the 0-- so I\Ndon't have to write that--
Dialogue: 0,0:03:00.32,0:03:03.96,Default,,0000,0000,0000,,plus 6 choose 1.
Dialogue: 0,0:03:03.96,0:03:05.88,Default,,0000,0000,0000,,And actually, we'll probably\Njust have to figure out the
Dialogue: 0,0:03:05.88,0:03:07.33,Default,,0000,0000,0000,,third term because that's\Nthe only time where
Dialogue: 0,0:03:07.33,0:03:09.38,Default,,0000,0000,0000,,they start to change.
Dialogue: 0,0:03:09.38,0:03:21.20,Default,,0000,0000,0000,,6 choose 1 times 1 to the fifth\Npower times 2x to the
Dialogue: 0,0:03:21.20,0:03:23.21,Default,,0000,0000,0000,,first power, times 2x.
Dialogue: 0,0:03:23.21,0:03:29.39,Default,,0000,0000,0000,,Plus 6 choose 2 times-- I mean\Nthese 1s, it doesn't matter
Dialogue: 0,0:03:29.39,0:03:37.20,Default,,0000,0000,0000,,what power it is-- 1 to the\Nfourth times 2x squared plus--
Dialogue: 0,0:03:37.20,0:03:40.33,Default,,0000,0000,0000,,let's see, they want the first\Nso let's do the last one-- 6
Dialogue: 0,0:03:40.33,0:03:47.40,Default,,0000,0000,0000,,choose 3 times 1 to the third--\Nyou have to decrement
Dialogue: 0,0:03:47.40,0:03:51.47,Default,,0000,0000,0000,,the 1 every time-- times\N2x to the third.
Dialogue: 0,0:03:51.47,0:03:53.18,Default,,0000,0000,0000,,All right, and if we look\Nat the choices,
Dialogue: 0,0:03:53.18,0:03:54.74,Default,,0000,0000,0000,,just to save time.
Dialogue: 0,0:03:54.74,0:03:57.56,Default,,0000,0000,0000,,If we look at a choices, OK, we\Nagree that the first term
Dialogue: 0,0:03:57.56,0:03:58.47,Default,,0000,0000,0000,,is going to be 1.
Dialogue: 0,0:03:58.47,0:04:01.22,Default,,0000,0000,0000,,So that, we can definitely say\Nis going to be 1 because all
Dialogue: 0,0:04:01.22,0:04:01.91,Default,,0000,0000,0000,,the choices are 1.
Dialogue: 0,0:04:01.91,0:04:04.64,Default,,0000,0000,0000,,The second term is definitely\Ngoing to be 12x.
Dialogue: 0,0:04:04.64,0:04:07.05,Default,,0000,0000,0000,,And then we start getting\Nsome disagreement
Dialogue: 0,0:04:07.05,0:04:08.08,Default,,0000,0000,0000,,on the third term.
Dialogue: 0,0:04:08.08,0:04:09.18,Default,,0000,0000,0000,,So let's see what\Nthe third term.
Dialogue: 0,0:04:09.18,0:04:14.46,Default,,0000,0000,0000,,6 choose to that's equal to 6\Nback toward ill over to affect
Dialogue: 0,0:04:14.46,0:04:21.70,Default,,0000,0000,0000,,or you over fix minus is too\Nfact or you scroll down a
Dialogue: 0,0:04:21.70,0:04:27.38,Default,,0000,0000,0000,,little bit so that is equal to\N6 factorial, which is 6 times
Dialogue: 0,0:04:27.38,0:04:30.77,Default,,0000,0000,0000,,5 times 4 times 3\Ntimes 2 times 1.
Dialogue: 0,0:04:30.77,0:04:32.73,Default,,0000,0000,0000,,But the 1 doesn't\Nchange anything.
Dialogue: 0,0:04:32.73,0:04:36.35,Default,,0000,0000,0000,,Divided by 2 factorial, which\Nis 2 times 1, or just 2.
Dialogue: 0,0:04:36.35,0:04:39.51,Default,,0000,0000,0000,,Divided by 6 minus 2 factorial,\Nthat's 4 factorial.
Dialogue: 0,0:04:39.51,0:04:44.53,Default,,0000,0000,0000,,Royal So times 4 times\N3 times 2 times 1.
Dialogue: 0,0:04:44.53,0:04:46.42,Default,,0000,0000,0000,,That cancels out with that.
Dialogue: 0,0:04:46.42,0:04:48.30,Default,,0000,0000,0000,,The 2 and the 6 cancel out.
Dialogue: 0,0:04:48.30,0:04:49.03,Default,,0000,0000,0000,,You get 3.
Dialogue: 0,0:04:49.03,0:04:51.83,Default,,0000,0000,0000,,So you end up with 3 times 5.
Dialogue: 0,0:04:51.83,0:04:59.20,Default,,0000,0000,0000,,So the 6 choose 2 becomes 15,\N3 times 5, that's 15.
Dialogue: 0,0:04:59.20,0:05:04.30,Default,,0000,0000,0000,,Times 1 to the fourth, which\Nis 1, times 2x squared.
Dialogue: 0,0:05:04.30,0:05:07.78,Default,,0000,0000,0000,,So times 4x squared.
Dialogue: 0,0:05:07.78,0:05:11.02,Default,,0000,0000,0000,,And so that's 15 times 4,\Nthat's 60x squared.
Dialogue: 0,0:05:11.02,0:05:14.72,Default,,0000,0000,0000,,So the third term is going\Nto be 60x squared.
Dialogue: 0,0:05:14.72,0:05:15.70,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:05:15.70,0:05:17.76,Default,,0000,0000,0000,,Choice D is the only one\Nthat has 60x squared
Dialogue: 0,0:05:17.76,0:05:18.58,Default,,0000,0000,0000,,as the third term.
Dialogue: 0,0:05:18.58,0:05:18.93,Default,,0000,0000,0000,,So we're done.
Dialogue: 0,0:05:18.93,0:05:21.12,Default,,0000,0000,0000,,We didn't even have to go\Nto the fourth term.
Dialogue: 0,0:05:21.12,0:05:23.77,Default,,0000,0000,0000,,So, by carefully looking at the\Nchoices we actually only
Dialogue: 0,0:05:23.77,0:05:27.12,Default,,0000,0000,0000,,have to evaluate one\Nof the terms.
Dialogue: 0,0:05:27.12,0:05:28.37,Default,,0000,0000,0000,,Next problem.
Dialogue: 0,0:05:28.37,0:05:31.94,Default,,0000,0000,0000,,
Dialogue: 0,0:05:31.94,0:05:33.23,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:05:33.23,0:05:33.78,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:05:33.78,0:05:34.83,Default,,0000,0000,0000,,Problem 69.
Dialogue: 0,0:05:34.83,0:05:46.75,Default,,0000,0000,0000,,They say-- let me copy and paste\Nit-- what is the sum of
Dialogue: 0,0:05:46.75,0:05:51.31,Default,,0000,0000,0000,,the infinite geometric series,\N1/2 plus 1/4 plus 1/8 all the
Dialogue: 0,0:05:51.31,0:05:51.94,Default,,0000,0000,0000,,way up there.
Dialogue: 0,0:05:51.94,0:05:52.99,Default,,0000,0000,0000,,I'll be frank.
Dialogue: 0,0:05:52.99,0:05:54.94,Default,,0000,0000,0000,,I do now remember the formula.
Dialogue: 0,0:05:54.94,0:05:58.03,Default,,0000,0000,0000,,But a lot of times in life you\Nmight forget the formula.
Dialogue: 0,0:05:58.03,0:06:00.05,Default,,0000,0000,0000,,And there's a trick\Nis to re-proving
Dialogue: 0,0:06:00.05,0:06:01.28,Default,,0000,0000,0000,,the formula for yourself.
Dialogue: 0,0:06:01.28,0:06:03.80,Default,,0000,0000,0000,,Just in case you forget\Nit when you're 32
Dialogue: 0,0:06:03.80,0:06:05.39,Default,,0000,0000,0000,,years old like me.
Dialogue: 0,0:06:05.39,0:06:08.70,Default,,0000,0000,0000,,So let's say that I'm trying to\Nfind the infinite sum of a
Dialogue: 0,0:06:08.70,0:06:09.69,Default,,0000,0000,0000,,geometric series.
Dialogue: 0,0:06:09.69,0:06:12.38,Default,,0000,0000,0000,,So let's say that the sum-- and\NI'm going to re-prove it
Dialogue: 0,0:06:12.38,0:06:14.80,Default,,0000,0000,0000,,right here in the midst of this\Ntest. And I've done this
Dialogue: 0,0:06:14.80,0:06:16.73,Default,,0000,0000,0000,,in midst of tests and\Nit's saved me.
Dialogue: 0,0:06:16.73,0:06:17.83,Default,,0000,0000,0000,,So it's nice to know\Nthe proof.
Dialogue: 0,0:06:17.83,0:06:20.42,Default,,0000,0000,0000,,And it actually impresses\Npeople, if you know people who
Dialogue: 0,0:06:20.42,0:06:21.93,Default,,0000,0000,0000,,are impressed by proofs.
Dialogue: 0,0:06:21.93,0:06:28.53,Default,,0000,0000,0000,,So let's say the sum is equal to\Na to the 0-- well actually
Dialogue: 0,0:06:28.53,0:06:30.93,Default,,0000,0000,0000,,let's say-- well, see\Nthis is interesting.
Dialogue: 0,0:06:30.93,0:06:32.37,Default,,0000,0000,0000,,Because they're starting\Nat 1/2.
Dialogue: 0,0:06:32.37,0:06:33.91,Default,,0000,0000,0000,,Actually let's just do\Nthe concrete numbers.
Dialogue: 0,0:06:33.91,0:06:36.15,Default,,0000,0000,0000,,Because we want to figure\Nout this exact sum.
Dialogue: 0,0:06:36.15,0:06:43.82,Default,,0000,0000,0000,,So let's say that the sum is\Nequal to 1/2 to the 1 1/2
Dialogue: 0,0:06:43.82,0:06:49.17,Default,,0000,0000,0000,,squared plus 1/2 to the third--\Nit's the whole 1/2 to
Dialogue: 0,0:06:49.17,0:06:53.36,Default,,0000,0000,0000,,the third-- plus-- and\Nyou just keep going.
Dialogue: 0,0:06:53.36,0:06:54.57,Default,,0000,0000,0000,,Fair enough.
Dialogue: 0,0:06:54.57,0:06:57.46,Default,,0000,0000,0000,,Now let's take 1/2\Ntimes the sum.
Dialogue: 0,0:06:57.46,0:07:01.31,Default,,0000,0000,0000,,So 1/2 times this.
Dialogue: 0,0:07:01.31,0:07:02.81,Default,,0000,0000,0000,,What is this equal to?
Dialogue: 0,0:07:02.81,0:07:06.50,Default,,0000,0000,0000,,Well if I took 1/2 times this\Nwhole thing, now 1/2, I would
Dialogue: 0,0:07:06.50,0:07:08.80,Default,,0000,0000,0000,,have to distribute it over each\Nof these terms. So 1/2
Dialogue: 0,0:07:08.80,0:07:12.35,Default,,0000,0000,0000,,times this term, times this\Nfirst term, 1/2 times 1/2 to
Dialogue: 0,0:07:12.35,0:07:14.51,Default,,0000,0000,0000,,the first, well that's now\Ngoing to be 1/2 squared.
Dialogue: 0,0:07:14.51,0:07:17.19,Default,,0000,0000,0000,,
Dialogue: 0,0:07:17.19,0:07:18.57,Default,,0000,0000,0000,,1/2 squared.
Dialogue: 0,0:07:18.57,0:07:21.29,Default,,0000,0000,0000,,All I did is I'm distributing\Nthis 1/2 times each of the
Dialogue: 0,0:07:21.29,0:07:24.59,Default,,0000,0000,0000,,terms. 1/2 times 1/2 to the\Nfirst. That's 1/2 squared.
Dialogue: 0,0:07:24.59,0:07:27.82,Default,,0000,0000,0000,,Now 1/2 times 1/2 squared, well\Nthat's going to be 1/2 to
Dialogue: 0,0:07:27.82,0:07:29.45,Default,,0000,0000,0000,,the third power.
Dialogue: 0,0:07:29.45,0:07:30.83,Default,,0000,0000,0000,,And it just keeps going.
Dialogue: 0,0:07:30.83,0:07:33.42,Default,,0000,0000,0000,,It's an infinite series.
Dialogue: 0,0:07:33.42,0:07:34.71,Default,,0000,0000,0000,,So let me ask you something.
Dialogue: 0,0:07:34.71,0:07:37.84,Default,,0000,0000,0000,,If I subtracted this from\Nthat, what do I get?
Dialogue: 0,0:07:37.84,0:07:39.90,Default,,0000,0000,0000,,Let's see what I get.
Dialogue: 0,0:07:39.90,0:07:45.77,Default,,0000,0000,0000,,So I get S minus 1/2 S-- I'm\Nsubtracting that from that--
Dialogue: 0,0:07:45.77,0:07:49.42,Default,,0000,0000,0000,,is equal to this minus this.
Dialogue: 0,0:07:49.42,0:07:51.49,Default,,0000,0000,0000,,Well if I'm subtracting the\Ngreen stuff from the yellow
Dialogue: 0,0:07:51.49,0:07:53.81,Default,,0000,0000,0000,,stuff, this is going to cancel\Nout with that, that's going to
Dialogue: 0,0:07:53.81,0:07:54.82,Default,,0000,0000,0000,,cancel out with that.
Dialogue: 0,0:07:54.82,0:07:55.98,Default,,0000,0000,0000,,And all I'm going to be\Nleft with is this
Dialogue: 0,0:07:55.98,0:07:59.21,Default,,0000,0000,0000,,first thing right here.
Dialogue: 0,0:07:59.21,0:07:59.90,Default,,0000,0000,0000,,That's the trick.
Dialogue: 0,0:07:59.90,0:08:01.85,Default,,0000,0000,0000,,And you could prove it generally\Nfor any base.
Dialogue: 0,0:08:01.85,0:08:02.98,Default,,0000,0000,0000,,And I thought it was interesting\Nbecause a lot of
Dialogue: 0,0:08:02.98,0:08:04.71,Default,,0000,0000,0000,,times they start with\Nthe 0th power.
Dialogue: 0,0:08:04.71,0:08:07.14,Default,,0000,0000,0000,,And this problem was interesting\Nbecause they start
Dialogue: 0,0:08:07.14,0:08:08.78,Default,,0000,0000,0000,,with 1/2 to the first power.
Dialogue: 0,0:08:08.78,0:08:10.16,Default,,0000,0000,0000,,So anyway let's figure it out.
Dialogue: 0,0:08:10.16,0:08:13.97,Default,,0000,0000,0000,,S minus 1/2 S, well\Nthat's just 1/2 S.
Dialogue: 0,0:08:13.97,0:08:17.81,Default,,0000,0000,0000,,1/2 S is equal to 1/2.
Dialogue: 0,0:08:17.81,0:08:20.38,Default,,0000,0000,0000,,And then you have S is-- divided\Nboth sides by 1/2 or
Dialogue: 0,0:08:20.38,0:08:25.52,Default,,0000,0000,0000,,multiply both sides by 2 and\Nyou get-- S is equal to 1.
Dialogue: 0,0:08:25.52,0:08:30.54,Default,,0000,0000,0000,,Now, a lot of you may know\Nthere is a formula.
Dialogue: 0,0:08:30.54,0:08:33.31,Default,,0000,0000,0000,,Let me tell you what\Nthe formula says.
Dialogue: 0,0:08:33.31,0:08:35.89,Default,,0000,0000,0000,,The formula that you may or may\Nnot have memorized, is if
Dialogue: 0,0:08:35.89,0:08:43.84,Default,,0000,0000,0000,,you start at n is equal to 0 and\Nif you go to infinity of a
Dialogue: 0,0:08:43.84,0:08:47.95,Default,,0000,0000,0000,,to the n-- in this case a is\N1/2-- this is going to be
Dialogue: 0,0:08:47.95,0:08:53.21,Default,,0000,0000,0000,,equal to 1 over 1 minus a.
Dialogue: 0,0:08:53.21,0:08:55.95,Default,,0000,0000,0000,,So your natural reaction would\Nhave been, oh OK, well that
Dialogue: 0,0:08:55.95,0:09:01.64,Default,,0000,0000,0000,,sum that we saw, that's equal to\Nthe sum of n is equal to 0
Dialogue: 0,0:09:01.64,0:09:06.86,Default,,0000,0000,0000,,to infinity of 1/2 to the n,\Nwhich would be equal to 1 over
Dialogue: 0,0:09:06.86,0:09:09.18,Default,,0000,0000,0000,,1 minus 1/2.
Dialogue: 0,0:09:09.18,0:09:09.67,Default,,0000,0000,0000,,What's that?
Dialogue: 0,0:09:09.67,0:09:13.13,Default,,0000,0000,0000,,1 minus 1, that's 1 over 1/2,\Nwhich would be equal to 2.
Dialogue: 0,0:09:13.13,0:09:16.20,Default,,0000,0000,0000,,And you'd be like, my God, what\Ndid I do wrong there?
Dialogue: 0,0:09:16.20,0:09:18.58,Default,,0000,0000,0000,,And I'll show you what\Nyou did wrong.
Dialogue: 0,0:09:18.58,0:09:21.11,Default,,0000,0000,0000,,This sum that you took the sum\Nof-- remember n is starting at
Dialogue: 0,0:09:21.11,0:09:27.60,Default,,0000,0000,0000,,0-- so this sum that sums up to\N2 is 1/2 to 0 is 1 plus 1/2
Dialogue: 0,0:09:27.60,0:09:35.03,Default,,0000,0000,0000,,to the 1, 1/2, plus 1/2 squared,\N1/4, plus 1/8, so on
Dialogue: 0,0:09:35.03,0:09:36.08,Default,,0000,0000,0000,,and so forth.
Dialogue: 0,0:09:36.08,0:09:37.38,Default,,0000,0000,0000,,That is equal to 2.
Dialogue: 0,0:09:37.38,0:09:40.30,Default,,0000,0000,0000,,
Dialogue: 0,0:09:40.30,0:09:42.78,Default,,0000,0000,0000,,This problem-- and that's why it\Nwas a little tricky-- this
Dialogue: 0,0:09:42.78,0:09:45.33,Default,,0000,0000,0000,,problem, they didn't want this\Nsum, they wanted this sum.
Dialogue: 0,0:09:45.33,0:09:52.02,Default,,0000,0000,0000,,1/2 plus 1/4 plus 1/8 plus,\Nso forth and so on.
Dialogue: 0,0:09:52.02,0:09:54.00,Default,,0000,0000,0000,,So they wanted everything\Nbut this 1.
Dialogue: 0,0:09:54.00,0:09:56.23,Default,,0000,0000,0000,,So if you subtracted this 1 from\Nboth sides then you would
Dialogue: 0,0:09:56.23,0:09:59.04,Default,,0000,0000,0000,,say, oh this sum must be equal\Nto 1, which is the answer we
Dialogue: 0,0:09:59.04,0:10:00.40,Default,,0000,0000,0000,,got that way.
Dialogue: 0,0:10:00.40,0:10:03.14,Default,,0000,0000,0000,,So that's why sometimes,\Nmemorizing formulas, this
Dialogue: 0,0:10:03.14,0:10:06.35,Default,,0000,0000,0000,,formula's a nice formula to\Nmemorize, it's very simple.
Dialogue: 0,0:10:06.35,0:10:08.43,Default,,0000,0000,0000,,But it sometimes becomes really\Nconfusing when you're
Dialogue: 0,0:10:08.43,0:10:11.63,Default,,0000,0000,0000,,like, oh boy, but this is when\Nn starts 0 but what happens
Dialogue: 0,0:10:11.63,0:10:12.81,Default,,0000,0000,0000,,when n starts at 1.
Dialogue: 0,0:10:12.81,0:10:14.90,Default,,0000,0000,0000,,And to some degree I really like\Nthis problem because it's
Dialogue: 0,0:10:14.90,0:10:16.94,Default,,0000,0000,0000,,testing you to see if you really\Nunderstand what the
Dialogue: 0,0:10:16.94,0:10:18.45,Default,,0000,0000,0000,,formula's all about.
Dialogue: 0,0:10:18.45,0:10:21.48,Default,,0000,0000,0000,,Anyway, I'll see you\Nin the next video.
Dialogue: 0,0:10:21.48,0:10:22.50,Default,,0000,0000,0000,,