Answer:
A.) 90 payments are necessary to pay off the loan amount assuming no deposit was made
B.) Final payment = 15377 dollars
Explanation:
Given that the first payment = $200
Paying at 5% increase every month. That is, in the next month couple will pay 200 × 1.05 and the month after, they will pay 200 × 1.05^2
This sequnce of payments is geometric progression.This means that payment made at month number (n+1) is
200 × 1.05^n
Sum of payments made in the first (n+1) month is equal to:
200 + 200 × 1.05 + 200 × 1.05^2 + ...+ 200 × 1.05^n
= 200 × (1 + 1.05 + 1.05^2 + ...+1.05^n)
= 200 × (1.05^{n+1}-1)/(1.05-1)
= 200/0.05 × (1.05^{n+1}-1)
= 4000 × (1.05^{n+1}-1
This sum should be equal or greater than the value of the house $318,921.46.
If n+1 = 89,
Sum of payments will be $303,544.25 which is not up to the value of the house
If n+1 = 90,
Sum of payments will be $318,921.460.
Therefore, 90 payments are necessary to pay off the loan amount assuming no deposit was made.
Assuming no deposit was made. Final payment will be
200 × 1.05^{89} = 15377.212
= 15377 dollars ( approximately)