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

if he wants to pay 1.2 times the amount of the previous month, then 1.2 becomes a multiplier from the current month's amount, to get the "next" month's amount, namely 1.2 is the "common ratio", that simply means is a geometric sequence, with a first term value of 200


`\bf \textit{sum of a finite geometric sequence}\\\\ \begin{array}{llll} S_n\implies \sum\limits_(n=1)^(30)\ a_1\cdot r^(n-1)\\\\ S_n=a_1\left( \cfrac{1-r^n}{1-r} \right) \end{array} \qquad \begin{cases} n=n^(th) \ term\\ a_1=\textit{value of first term}\\ r=\textit{common ratio}\\ ----------\\ r=1.2\\ a_1=200 \end{cases} \\\\\\ \sum\limits_(n=1)^(30)\ 200(1.2)^(n-1)\iff 200\left( \cfrac{1-1.2^(30)}{1-1.2} \right)`

is convergent, because is a finite sum

Answer 2

Answer: The series is divergent.

Explanation:

The geometric series is given by

`\sum_(n=0)^(\infty )ar^n=a+ar+ar^2+...`

• If |r| < 1 then the geometric series converges to.

• If |r| ≥1 then the geometric series diverges.

The given sentences a geometric series,

where the first payment represents the first term a.

The number of times he wants to pay back for succeeding payments represent the common ration i.e. r=1.2.

The first 30 payments represents the number of terms to use in finding the sumi.e. n=30.

Since the common ratio, r>1, the series is divergent.