Question

Tom 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 as the amount he paid the month before. Explain to Tom 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

This is a geometric series. The first payment represents the first term, a. The number of times he wants to pay back for subsequent payments represent the common ration, r. The first 30 payments represents the number of terms to use in finding the sum, n.

Hence, a = $ 200, n = 30 and r = 1.2.

The sum of a geometric sequence is given as:

Sn = (a((r^n)-1))/(r-1)

=(200((1.2^30)-1))/(1.2-1)

= $ 236, 376. 31

Explain why this series is convergent or divergent.

Since the common ratio, r, is greater than 1, the series is divergent

Answer 2

In this geometric series, a represents the first term, what he pays back monthly is the common ratio which is 1.2, and the first 30 payments are what is being used to find the total amount which is represented by n. Thus, a=$200, n=30 and r=1.2. The sum is found by sn=(a((r^n)-1))/(r-1), which will be 200((1.2^30)-1))/(1.2-1). The solution is = $ 236,376 and the series is divergent because of the common ratio being greater than 1.

Explanation: