Word Problem

[jmm811]

I have a word problem - I know I need to you a ^n+2 equation but I am not sure how to solve.

A man borrowed $580. He made his first payment of $10. He made a $10 payment today. He would like to make monthly payments - each $2 more than the last until the loan is paid off. Assuming no interest - how many months will it take to repay the loan.

Thanks for your help!

 

[soroban]

Hello, jmm811!

The wording is misleading . . . It took a while to figure it out.

A man borrowed $580. .He made his first payment of $10 today.
He would like to make monthly payments, each $2 more than the last, until the loan is paid off.
Assuming no interest, how many months will it take to repay the loan?

We have: \(\displaystyle \:10\,+\,12\,+\,14\,=\,16\,+\,\cdots\,+\,(8\,+\,2n) \;=\;580\)

The left side is an arithmetic series with first term \(\displaystyle a\,=\,10\) and common difference \(\displaystyle d\,=\,2\)

Its sum is: \(\displaystyle \:\frac{n}{2}\left[2(10)\,+\,2(n\,-\,1)\right] \:=\:580\)

This simplifies to the quadratic: \(\displaystyle \:n^2\,+\,9n\,-\,580\:=\:0\)

. . which factors: \(\displaystyle \n\,-\,20)(n\,+\,29)\:=\:0\)

. . and has the positive root: \(\displaystyle \,n\,=\,20\;\) [20 payments are required.]


He has already made the first payment.
. . Therefore, it will take him 19 more months to repay the loan.