so he pays 5% straight away , so he has 0.95 x 35000 left to pay = 33250 nd ill call this P to make the work easiesr, and let R be the repayment
now the rate = 12%pa= 1%/month = 0.01/month
and A= P ( 1+R)^n
now let A (n) be the amount owing after n months
so A (1) = P ( 1.01) -R { he is charged interest then makes a payment }
A (2) = [ P (1.01) -R] (1.01) -R { same idea as above}
A (2) = P (1.01)^2 -R - R(1.01)
A (2) = P ( 1.01)^2 -R (1+1.01) { pattern is easy to see, in the last bracket we will have n terms, with the nth term having a power of n-1 }
so A (n) = P (1.01)^n - R ( 1+1.01 + 1.01^2 +.... 1.01^(n-1) )
now apply the sum of a GP formula to last part, with a=1, r=1.01, n=n
A (n) = P (1.01)^n -R [ 1 ( 1-(1.01)^n ) / (1-1.01) ]
A (n) = P (1.01)^n +100R [ 1- (1.01)^n]
now in 4yrs ( 48months , he owes nothin)
A ( 48) = 0
so P(1.01)^48 + 100R [ 1- (1.01)^48] =0
now solve for R
100R [ 1 -(1.01)^48] = - P ( 1.01)^48
100R= -P(1.01)^48 / [ ( 1-(1.01)^48 ]
R = -P ( 1.01)^48 / [ 100 ( 1-(1.01)^48 ) ]
and sub in P
R= $ 875.60
now he paided this 48 times
so he ends up paying $875.60 x 48 =$42 028.80
hmm not quite, you have fallen into the trap.
remember, we have ignored for the moment the 5% that paided as the desposit,
add this 5% on ie 0.05 x 35000 = $1750
so final answer = 1750 + 42 028. 80 and the book is correct