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?

1 Answer
Nov 29, 2016

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

p_0=20$p0=20$
p_k=20$+k* 15$pk=20$+k15$ 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.