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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 4: Finite geometric series

# 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? • 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.
• if you keep on multiplying 1 by 10/11 for an infinite amount of times, would it eventually become zero? Thanks in advance! • this thing is sooo confusing • At , it says a sub i is equal to a sub i minus i times 9/10. Is this supposed to be i-1? • 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! • 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 at • 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.
• Wait at , I thought dividing by 1/10 was multiplying by 10 not 100?? • 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.
• at , he said 80 but shouldn't it be 78 because of n-1? • 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. • 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.