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## Get ready for AP® Calculus

### Unit 9: Lesson 5

Geometric series- Geometric series introduction
- Finite geometric series formula
- Worked examples: finite geometric series
- Geometric series formula
- Geometric series word problems: swing
- Geometric series word problems: hike
- Finite geometric series word problems
- Geometric series with sigma notation
- Worked example: finite geometric series (sigma notation)
- Finite geometric series

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# Worked examples: finite geometric series

Sal evaluates three geometric series (defined in various ways) using the finite geometric series formula a(1-rⁿ)/(1-r).

## Want to join the conversation?

- for the last problem wouldnt it be to the 31rst power?(12 votes)
- Good question. It's a little complicated, but:

The formula for the sum was made for a sequence of n terms. If there are 30 terms (n=30), and if the 1st term is 10, you multiply by 9/10 from i=2 to i=30 (29 terms - so you have (9/10)^29) plus 1 more for i=1 (the first term, 10) makes 30.

So if he had set it up like the earlier problem "10 + 10(9/10) + 10(9/10)^2 + ... + 10*(9/10)^29" then you add 1 to the 29. But when he just says "there's 30 terms" or "29 terms", you don't add anything.(9 votes)

- if you keep on multiplying 1 by 10/11 for an
**infinite**amount of times, would it eventually become*zero*? Thanks in advance!(8 votes)- That is the correct intuition to have. Obviously, we don't reach 0 after any finite number of multiplications, but we say the product
*tends to 0*or*approaches 0*.(12 votes)

- Can someone help me understand how to find the first term when given the common ration and sum of the first n terms? I'm having trouble with the solving part. I get to the equation, but then it gets complicated trying to find a with fractions and exponents in the equation. Thanks in advance!(6 votes)
- a=S(1-r)/(1-r^n).(S-sum till n terms, n-number of terms, r-common ratio,a-first term)

If you have to find the first term given the sum, and common ratio, substitute the value of S, r and n and do the calculations carefully. You will get the value of a with this method.(5 votes)

- At4:55, it says a sub i is equal to a sub i minus i times 9/10. Is this supposed to be i-1?(4 votes)
- It's confusing... the dot is actually the multiplication symbol, not a dot on a "i".

So, Sal does have "i-1"(8 votes)

- When Sal simplifies 1(1- -.99^80) why does he simplify this portion to 1-.99^80 instead of 1+.99^80? It's written at4:27(3 votes)
- You may have forgotten to look at the parenthesis. As Sal writes it, it is (1 - (-0.99)^80). Since 80 is an even exponent, both -0.99 and +0.99 will output to the same number, so you can effectively cancel out the second negative sign, giving you 1 - 0.99^8. Since exponentiation happens before addition, we can do this and canceling out the two negative signs would be wrong.(5 votes)

- Wait at6:20, I thought dividing by 1/10 was multiplying by 10 not 100??(4 votes)
- it is, but the numerator is already being multiplied b a 10. so 10*10 = 100. Or shown more explicitly, 10(1-(9/10)^30)/(1/10) = 10 * 10(1-(9/10)^30) = 100(1-(9/10)^30).(2 votes)

- this thing is sooo confusing(4 votes)
- at2:53, he said 80 but shouldn't it be 78 because of n-1?(3 votes)
- I think you meant 79, but don't be confused:

The reason why he said there are 80 terms is because he included the first term, "1", and then up to the 79th exponent.

n = number of terms

You don't have to worry about n-1.(2 votes)

- Why is dividing by 1/11 the same as multiplying the numerator by 11?(1 vote)
- Hello Jyotika,

The answer is that when you divide by a fraction, the number gets larger. For instance, when you divide 1 by 1/4, you get 4, because 1/4 goes into 1, 4 times. Now, when you do the same thing, multiply 1 times 4, you get four. This is because you are doing an inverse operation on an inverse number, essentially doing the same thing with both, and as such, getting the same number. Hopefully this helps you.

Have fun doing your math, and I hope you succeed in your endeavors.(6 votes)

- I set this formula up right but always get a different answer. Like, for example, in one problem, the common ratio was 0.75 and the first term was 64. Also, the (n) factor was 4. I set it up according to the finite geometric series formula and got= 5.5. This is nowhere near the right answer, which is 175.(2 votes)
- With the first term 64 and the common ratio 0.75, clearly all four terms are positive and one term is 64. So the sum of the terms must be greater than 64, which should tell you immediately (without looking at the answer key) that 5.5 is clearly too low! It's a good habit to check whether or not your answers make sense.

Without seeing your steps, I cannot tell where you went wrong. Here are two possible correct methods:

1) Using the finite geometric series formula and converting 0.75 to 3/4, we find that the sum is

64[1 - (3/4)^4]/(1 - 3/4)

= 64(1 - 81/256)/(1/4)

= 64(175/256)/(1/4)

= (175/4)/(1/4)

= 175.

Try comparing what you did versus my solution using the finite geometric series formula, so that you can see where you went wrong.

2) We can find each term and then add them; this is not that hard because there are only four terms to add. It is helpful to convert 0.75 to 3/4.

The first term is 64.

The second term is 64(3/4) = 48.

The third term is 48(3/4) = 36.

The fourth term is 36(3/4) = 27.

The sum of the four terms is 64 + 48 + 36 + 27 = 175.

Note that using two different methods is also a good way to check your answer.(3 votes)

## Video transcript

- [Voiceover] We're asked to find the sum of the first 50 terms of this series, and you might immediately recognize it is a geometric series. When we go from one term to
the next, what are we doing? Well, we're multiplying by
10/11, to go from one to 10/11, you multiply by 10 over 11, then you multiply by 10 over 11 again, and we keep doing this, and we wanna find the
first 50 terms of it. So we can apply the formula we derived for the sum of a finite geometric series and that tells us that the sum of, let's say in this case the first 50 terms, actually let me do it down here, so the sum of the first 50 terms is going to be equal to the
first term, which is one, so it's gonna be one times one minus, let me do that in a different color, one times one minus the common ratio, so the common ratio here is
10/11, 10/11 to the 50th power, to the power of how many terms we have, all of that over one
minus our common ratio. And so I'm not gonna solve it completely, but we can simplify this a little bit, this is gonna be one minus, let me put parenthesis
here just to make sure we're not just taking
10 to the 50th power. So one minus 10/11 to the 50th power over, this is 11/11 minus 10/11 is one over 11, and so this is the same thing as multiplying the numerator by 11, and so this is gonna be equal to 11 times one minus 10/11 to the 50thpower, and you can try to
simplify this even more, but this gets up pretty far, at this point it is just arithmetic. Let's do another one of
these, this is kinda fun. So this is more clearly
a geometric series, and let's just first
think about how many terms we're gonna take the sum of. You might be tempted to say "Okay I'm gonna take it to the 79th power, "there must be 79 terms
here", but be very careful, because the first term is
when we're taking things to the zeroth power, we're
taking 0.99 to the zeroth power. The second term is where we're
taking it to the first power, the third term is where we're
taking it to the second power, the fourth term is where we're
taking it to the third power so on and so forth, so this
right over here is the 80th, the 80th term, 80th term. So we wanna find S sub 80, and
so this is gonna be equal to our first term is gonna be one times one minus our common
ratio to the 80th power, to the 80th power, all over,
and I'm leaving a blank because we still need to
figure out our common ratio, all over one minus our common ratio. So at first you might say well maybe the common ratio here is 0.99, but notice we have a change in sign here, and the key thing is to say
well to go from term to the next what are we multiplying by? Well to go from the first
term to the second term, we multiply by negative 0.99. And then, so we're
multiplying by negative 0.99. Now to go to the next term, we're again multiplying by negative 0.99, so the common ratio is not positive 0.99, but negative zero, negative 0.99, so let me write that, negative 0.99, and of course that is going
to be to the 80th power, all over one minus negative 0.99. And so we could simplify
this a little bit, this is all going to be equal to, oh that one we don't have to
worry too much about that, and so this is going to be one minus, so negative 0.99 to the 80th power, I should put parenthesis
there to make sure we are taking the negative
0.99 to the 80th power. Well, we're taking it to an even power, so it's going to be positive,
so that's going to be the same thing as 0.99 to the 80th power, and all of that over, well
subtracting a negative that's just gonna be adding the positive, so all of that over 1.99, and we could attempt to
simplify it more but, if we had a calculator we could actually find this exact
value or close value actually, most calculators don't
give you the exact value when you take something to the 80th power, but this is what that sum is going to be. Let's do one more of these. Alright, so here we have a
series defined recursively and so it's useful to just think about what it would actually look like. So the first term is 10,
and then the next term, so the second term A sub two is equal to A sub one times 9/10, alright. So the next term is gonna be
the previous term times 9/10, so it's gonna be 10 times nine over 10, and then the next term
is gonna be that times, is gonna be the second term, the third term is the
second term times 9/10, so 10 times nine over
10, nine over 10 squared. And the way it's written right now, we don't have it written as
a finite geometric series, so let's say we wanna take the sum, let's say we want the sum of first, first I don't know 30 terms, sum of first 30 terms. So what will this be? Well we're gonna take S sub
one, S sub 30, oh I wrote ten, S sub 30, the sum of the first 30 terms, is gonna be equal to the first
term, we've done this before, the first term times one
minus the common ratio, one minus the common
ratio to the 30th power, all of that over one
minus the common ratio. And let's see we could, one minus 9/10, this is 1/10 right over
here, you divide by 1/10, this is the same thing
as multiplying by 100, so this is gonna be 100
times one minus 9/10 to the, oh let me write it this
way, 9/10 to the 30th power. And, actually these
parenthesis you always wanna put parenthesis there to make sure we see we're taking both the nine and the 10 or the 9/10, the whole
thing to the 30th power, not just the nine, so
there you go, did I.. yep there you go we're done.