A couple decides to buy a house which is currently valued at $318,921.46 on loan. The couple is willing to start paying $200.00 per month and are willing to increase their payment at a rate of 5% every month. How many payments are necessary to pay off the loan amount assuming no deposit was made (answer to the nearest whole number) ? What is the value of the final payment that they would make assuming no deposit was made (answer to the nearest whole number) ? [Hint: apply geometric sequences].

A.) 90 payments are necessary to pay off the loan amount assuming no deposit was made

B.) Final payment = 15377 dollars

Step-by-step 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)