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31 August, 16:21

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].

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  1. 31 August, 19:16
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    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)
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