Question

sigma notation and convergent/divergent series help

Im really confused with this problem :(
I dont understand how to do it at all. Its an algebra 2 question btw
Alex has taken out a loan for college. He started paying off the loan with a first payment of $200. Each month he pays, he wants to pay back 1.2 times the amount he paid the month before. Explain to Alex how to represent his first 30 payments in sigma notation. Then explain how to find the sum of his first 30 payments, using complete sentences. Explain why this series is convergent or divergent.

Answer 1

P1: 200
P2: 200*1.2
P3: 200*1.2^2
P4: 200*1.2^3
P5: 200*1.2^4
...
P30: 200*1.2^29

Can you add them up?

Answer 2

Add each one together separately? like (200)+(200*1.2)+(200*1.2^2)...etc?

Answer 3

There's some algebra in there.

200(1 + 1.2 + 1.2^2 + 1.2^3 + ... + 1.2^2.9)

Can you add up the Finite Geometric Series?

Answer 4

Im not sure what you mean by that sorry :S
Im supposed to turn it into sigma notation. I can manually solve it to get the sum of his first 30 payments, but I dont think im supposed to do that for the answer. Although correct me if im wrong, i dont really understand this stuff that well :(

Answer 5

The sigma notation is trivial. \(200\cdot \sum\limits_{n=0}^{29}1.2^{n}\)

The question remains, can you add up that Finite Geometric Series that is represented by the expression (whether it be written out or in sigma notation).

Answer 6

sn = n(t_1 + t_n)/2?

Answer 7

oh sorry, that was arithmetic series sum, not geometric

Answer 8

so i have to find the common ratio i think?

Answer 9

Oh, you'll have to do better than that. It should be obvious that the common ratio is 1.2. It's the only number in there!

1 + 1.2 + 1.2^2 + ... + 1.2^29 = S
1.2 + 1.2^2 + ... + 1.2^29 + 1.2^30 = S*1.2

(1 - 1.2^30)(1-1.2) = S

\(200\cdot \dfrac{1 - 1.2^{30}}{1-1.2}\)

It is a very important process that requires practice until it happens very, very fluently in your head.

Answer 10

OH, i didnt realize that 1.2 was the common ratio XD

Answer 11

so now i would plug the values into the formula given on http://home.windstream.net/okrebs/page133.html ?

Answer 12

for the sum of a finite geometric series

Answer 13

wait nvm i think i understand now, thanks!