Question

A man repays a loan of $3250 by paying $20 in the first month and then increases the payment by $15 every month. How long will it take him to clear the loan?

Answer 1

Define `p_k` as the payment in the month `k+1`.
We have:

`p_0=20$`
`p_k=20$+k* 15$` for `k=1,2,...`

At the end of the n-th month the total payment is:

`P_(n-1) = 20+ sum_1^(n-1)p_k = 20 + sum_1^(n-1) (20+15k)= 20n +15 sum_1^(n-1) k`

Using Gauss' formula for the sum of the first `(n-1)` integers:

`P_(n-1) = 20n + 15frac (n(n-1))2`

express this as an equation in `n` and pose `P_n=3250$`

`3250 = 20n + 15frac (n(n-1))2`

Solve for `n`:

`15n^2-15n+40n-6500=0`

`15n^2+25n-6500=0`

`n= frac (-25+- sqrt(625+4* 15 * 6500)) 30 = (-25+-625)/30`

Obviously we discard the negative solution and

`n=600/30=20`

So the debt is repaid at the end of the 20-th month.