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?

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Answer 1 · 2016-11-29T20:08:35

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.