Question

Margarita borrows $14,000 from her uncle and agrees to repay it in monthly installments of $400. Her uncle charges 0.9% interest per month on the balance.

(a) If her balance An in the nth month is given recursively by A0 = 14,000 and An = k · An − 1 − 400, what is k?

(b) Find her balance after two months. (Round your answer to the nearest cent.)

Answer 1

Explanation:

(a) We know that the balance An in the nth month is given by:

An = k · An-1 - 400

where A0 = 14,000. Substituting n = 1, we get:

A1 = k · A0 - 400

A1 = k · 14,000 - 400

We also know that her uncle charges 0.9% interest per month on the balance. This means that the balance after one month should be:

A1' = A0 + 0.009 · A0 - 400

A1' = 14,000 + 0.009 · 14,000 - 400

A1' = 14,126

Setting A1 = A1', we can solve for k:

k · 14,000 - 400 = 14,126

k = 1.009

Therefore, k is approximately 1.009.

(b) To find her balance after two months, we can use the recursive formula with n = 2:

A2 = k · A1 - 400

A2 = 1.009 · 14,126 - 400

Using a calculator, we get:

A2 ≈ 14,272.34

Therefore, her balance after two months is approximately $14,272.34.