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 the amount he paid the month before. Explain to Tom how to represent his first 30 payments in sequence notation. Then explain how to find the sum of his first 30 payments, using complete sentences.

Answer 1

his first payment will be 200 bucks, a₁ = 200, and he'd like to pay 1.2 times more on the next month, namely 1.2 is a multiplier or the "common ratio", thus his 30 payments are


`\bf \qquad \qquad \textit{sum of a finite geometric sequence}\\\\ S_n=\sum\limits_(i=1)^(n)\ a_1\cdot r^(i-1)\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ n=30 \end{cases} \\\\\\ S_(30)=a_1\left( \cfrac{1-r^(30)}{1-r} \right)\impliedby \textit{first 30 payments}`


`\bf S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ n=30\\ a_1=200\\ r=1.2 \end{cases} \\\\\\ S_(30)=200\left(\cfrac{1-1.2^(30)}{1-1.2} \right)\impliedby \textit{sum of the first 30 payments}`