Question

Suppose someone takes out a home improvement loan for $30000. The annual interest on the loan is 6% and is compounded monthly. The monthly payment is $600. Let a denote the amount owed at the end of the nth month. The payments start in the first month and are due the last day of every month - this means in a given month interest is added first, and then the payment is applied to the resulting loan Balance.

a. Give a recursive definition for a,. including the recurrence relation and the base case.
b. Suppose that the borrower would like a lower monthly payment. How large does the monthly payment need to be to ensure that the amount owed decreases every month?

Answer 1

a) Initial (Base) a
`_0` = 30000

b) the monthly payment needs to $150

Explanation:

Given the data in the question;

a. Give a recursive definition for a,. including the recurrence relation and the base case.

the annual interest on the loan is 6% compounded monthly;

⇒ i = 6% / 12

i = 0.5%

so, the recurrence relation is a
`_n` = a
`_(n-1)`( 1 + i/100) - 600

Here Initial (Base) a
`_0` = 30000

b) Suppose that the borrower would like a lower monthly payment. How large does the monthly payment need to be to ensure that the amount owed decreases every month

Let p be the required monthly payment,

then the condition will be; a
`_n` ≤ a
`_(n-1)`

a
`_(n-1)`( 1 + i/100) - p ≤ a
`_(n-1)`

a
`_(n-1)`( 1 + i/100) - a
`_(n-1)` ≤ p

a
`_(n-1)`( 1 + i/100 - 1) ≤ p

a
`_(n-1)`( i/100 ) ≤ p

a
`_(n-1)` ≤ p ( 100/i )

a
`_(1-1)` ≤ p ( 100/0.5 )

a
`_0` ≤ p (200)

we know that; a
`_0` = 30000

so

30000 ≤ p (200)

p ≤ 30000 / 200

p ≤ 150

Therefore, the monthly payment needs to $150