Algebra II: Binomial Expansions, Geometric Series Sum 67-69, binomial expansions and the sum of a geometric series
Algebra II: Binomial Expansions, Geometric Series Sum
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- We're on problem 67.
- They ask us how many terms of the binomial expansion of x
- squared plus 2y to the third to the
- twentieth power contain?
- And so right when you see, oh my god, to the twentieth
- power, that will take me forever to do, and then I'll
- have to count the terms. But remember, they're just asking
- you how many terms. They're not asking
- you to find the expansion.
- So you just have to say, oh boy, let me just think about
- this a little bit.
- And think of it this way: x plus y to the first power has
- how many terms?
- It has-- well it's just x plus y.
- And so it has 2 terms. x plus y, squared.
- That is equal to x squared plus 2xy plus y squared.
- So that has 3 terms. If I were to do x plus e The third
- power, I could do a binomial expansion, I could do it with
- Pascal's Triangle.
- Let me do it with Pascal's Triangle.
- I have the 1 and the 1.
- That's to the first power.
- The second power, those are the coefficients.
- The third power, the coefficients become 1, 3--
- because 1 plus 2 is 3-- 3, 1.
- And actually I don't even have to expand it out.
- Ill do it just for a little practice.
- It x to the third plus 3x squared y plus 3xy squared
- plus y to the third.
- I just used these.
- But when you take any binomial and you take it to the third
- power, you end up with 4 terms. And depending on how
- you think of it, if you think of it with Pascal's Triangle,
- every time we go down we're just going to be adding
- another term.
- When we take the fourth power it's going to have 5 terms. Or
- when you think about, when we use the binomial expansion,
- and you might want to watch some videos on that.
- There you also have, if you're taking it to the nth power,
- you end up with n plus 1 terms.
- So in this example, we have a binomial-- a binomial just
- means a two-term polynomial.
- And we're taking it to the twentieth power.
- If we were taking it to third power, we'd have four terms.
- If we were taking it to the fourth power, we'd have 5
- terms. So if we're taking it to the twentieth power, we're
- going to have 21 terms. And that's choice B.
- Next problem.
- All right, they want us to keep doing these and we'll do
- it because they want us to.
- All right, what are the first four terms of the expansion of
- 1 plus 2x to the sixth power?
- And here we'll just use the binomial expansion because I
- think that's what they want us to do.
- And that's probably the fastest way to do it.
- So let's just think of the coefficients first. So this is
- going to be-- well let's just do it.
- This is going to be equal to 6 choose 0.
- We're just going to do the first four terms. Times 1 to
- the 6 times 2x to the 0-- so I don't have to write that--
- plus 6 choose 1.
- And actually, we'll probably just have to figure out the
- third term because that's the only time where
- they start to change.
- 6 choose 1 times 1 to the fifth power times 2x to the
- first power, times 2x.
- Plus 6 choose 2 times-- I mean these 1s, it doesn't matter
- what power it is-- 1 to the fourth times 2x squared plus--
- let's see, they want the first so let's do the last one-- 6
- choose 3 times 1 to the third-- you have to decrement
- the 1 every time-- times 2x to the third.
- All right, and if we look at the choices,
- just to save time.
- If we look at a choices, OK, we agree that the first term
- is going to be 1.
- So that, we can definitely say is going to be 1 because all
- the choices are 1.
- The second term is definitely going to be 12x.
- And then we start getting some disagreement
- on the third term.
- So let's see what the third term.
- 6 choose to that's equal to 6 back toward ill over to affect
- or you over fix minus is too fact or you scroll down a
- little bit so that is equal to 6 factorial, which is 6 times
- 5 times 4 times 3 times 2 times 1.
- But the 1 doesn't change anything.
- Divided by 2 factorial, which is 2 times 1, or just 2.
- Divided by 6 minus 2 factorial, that's 4 factorial.
- Royal So times 4 times 3 times 2 times 1.
- That cancels out with that.
- The 2 and the 6 cancel out.
- You get 3.
- So you end up with 3 times 5.
- So the 6 choose 2 becomes 15, 3 times 5, that's 15.
- Times 1 to the fourth, which is 1, times 2x squared.
- So times 4x squared.
- And so that's 15 times 4, that's 60x squared.
- So the third term is going to be 60x squared.
- And we're done.
- Choice D is the only one that has 60x squared
- as the third term.
- So we're done.
- We didn't even have to go to the fourth term.
- So, by carefully looking at the choices we actually only
- have to evaluate one of the terms.
- Next problem.
- Let's see.
- All right.
- Problem 69.
- They say-- let me copy and paste it-- what is the sum of
- the infinite geometric series, 1/2 plus 1/4 plus 1/8 all the
- way up there.
- I'll be frank.
- I do now remember the formula.
- But a lot of times in life you might forget the formula.
- And there's a trick is to re-proving
- the formula for yourself.
- Just in case you forget it when you're 32
- years old like me.
- So let's say that I'm trying to find the infinite sum of a
- geometric series.
- So let's say that the sum-- and I'm going to re-prove it
- right here in the midst of this test. And I've done this
- in midst of tests and it's saved me.
- So it's nice to know the proof.
- And it actually impresses people, if you know people who
- are impressed by proofs.
- So let's say the sum is equal to a to the 0-- well actually
- let's say-- well, see this is interesting.
- Because they're starting at 1/2.
- Actually let's just do the concrete numbers.
- Because we want to figure out this exact sum.
- So let's say that the sum is equal to 1/2 to the 1 1/2
- squared plus 1/2 to the third-- it's the whole 1/2 to
- the third-- plus-- and you just keep going.
- Fair enough.
- Now let's take 1/2 times the sum.
- So 1/2 times this.
- What is this equal to?
- Well if I took 1/2 times this whole thing, now 1/2, I would
- have to distribute it over each of these terms. So 1/2
- times this term, times this first term, 1/2 times 1/2 to
- the first, well that's now going to be 1/2 squared.
- 1/2 squared.
- All I did is I'm distributing this 1/2 times each of the
- terms. 1/2 times 1/2 to the first. That's 1/2 squared.
- Now 1/2 times 1/2 squared, well that's going to be 1/2 to
- the third power.
- And it just keeps going.
- It's an infinite series.
- So let me ask you something.
- If I subtracted this from that, what do I get?
- Let's see what I get.
- So I get S minus 1/2 S-- I'm subtracting that from that--
- is equal to this minus this.
- Well if I'm subtracting the green stuff from the yellow
- stuff, this is going to cancel out with that, that's going to
- cancel out with that.
- And all I'm going to be left with is this
- first thing right here.
- That's the trick.
- And you could prove it generally for any base.
- And I thought it was interesting because a lot of
- times they start with the 0th power.
- And this problem was interesting because they start
- with 1/2 to the first power.
- So anyway let's figure it out.
- S minus 1/2 S, well that's just 1/2 S.
- 1/2 S is equal to 1/2.
- And then you have S is-- divided both sides by 1/2 or
- multiply both sides by 2 and you get-- S is equal to 1.
- Now, a lot of you may know there is a formula.
- Let me tell you what the formula says.
- The formula that you may or may not have memorized, is if
- you start at n is equal to 0 and if you go to infinity of a
- to the n-- in this case a is 1/2-- this is going to be
- equal to 1 over 1 minus a.
- So your natural reaction would have been, oh OK, well that
- sum that we saw, that's equal to the sum of n is equal to 0
- to infinity of 1/2 to the n, which would be equal to 1 over
- 1 minus 1/2.
- What's that?
- 1 minus 1, that's 1 over 1/2, which would be equal to 2.
- And you'd be like, my God, what did I do wrong there?
- And I'll show you what you did wrong.
- This sum that you took the sum of-- remember n is starting at
- 0-- so this sum that sums up to 2 is 1/2 to 0 is 1 plus 1/2
- to the 1, 1/2, plus 1/2 squared, 1/4, plus 1/8, so on
- and so forth.
- That is equal to 2.
- This problem-- and that's why it was a little tricky-- this
- problem, they didn't want this sum, they wanted this sum.
- 1/2 plus 1/4 plus 1/8 plus, so forth and so on.
- So they wanted everything but this 1.
- So if you subtracted this 1 from both sides then you would
- say, oh this sum must be equal to 1, which is the answer we
- got that way.
- So that's why sometimes, memorizing formulas, this
- formula's a nice formula to memorize, it's very simple.
- But it sometimes becomes really confusing when you're
- like, oh boy, but this is when n starts 0 but what happens
- when n starts at 1.
- And to some degree I really like this problem because it's
- testing you to see if you really understand what the
- formula's all about.
- Anyway, I'll see you in the next video.
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