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a) Every 400 can be considered as a single loan.

Loan = 400 * (interest rate) ^ (years) = 400 * (interest rate) ^ (months/12) = 400 * x ^ (n/12)

The whole loan is sum of these. The first loan is n months old, the second n-1 etc.

Amount of loan after n months = 400 * x ^ (n/12) + 400 * x ^ ((n-1)/12) + ... + 400 * x ^ (2/12) + 400 * x ^ (1/12)

In other words, the loan is sum 400 * x ^ (i/12) from i = 1 to i = n

Ratio ( 400 * x ^ (n/12) ) / ( 400 * x ^ ((n-1)/12) ) = x ^ (n/12 - (n-1)/12) = x ^ (1/12) doesn't depend on n, so we have a geometric progression

We can use its formula for the sum now:

sum = ( 400 * x ^ (1/12) ) * ( 1 - ( x ^ (1/12) )^n ) / ( 1 - x ^ (1/12) )

This is the amount of loan after n months. After n months the student has received n * 400, so the costs after n months are

costs = ( 400 * x ^ (1/12) ) * ( 1 - ( x ^ (1/12) )^n ) / ( 1 - x ^ (1/12) ) - n * 400

b)

0.4 * (400n - 2 500) is how much they will pay for me. To make profit 400n has to be > 2 500. n has to be at least 7.

If n is 7, they will pay 0.4 * (400 * 7 - 2 500) = 120

I will only profit if this is more than what the loan cost me. The costs after 7 months (n=7, x=1.01):

costs = ( 400 * 1.01 ^ (1/12) ) * ( 1 - ( 1.01 ^ (1/12) )^7 ) / ( 1 - 1.01 ^ (1/12) ) - 7 * 400
costs ~ 9.30626... ~ 9.3

So I will profit as long as n is at least 7.

If n is 50, they will pay 0.4 * (400 * 50 - 2 500) = 7 000

costs = ( 400 * 1.01 ^ (1/12) ) * ( 1 - ( 1.01 ^ (1/12) )^50 ) / ( 1 - 1.01 ^ (1/12) ) - 50 * 400
costs ~ 428.854... ~ 429

I will profit 7 000 - 429 = 6 571 !!! WOOT!