Margarita borrows $13,000 from her uncle and agrees to repay it in monthly installments of $800. Her uncle charges 0.2% interest per month on the balance.

Question

(a) If her balance An in the nth month is given recursively by A0 = 13,000 and An = k · An − 1 − 800, what is k?
k =

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

Answer 1

(a) To find k, we need to substitute the given values into the recursive formula and solve for k.
A0 = 13,000
A1 = k · A0 - 800

Substituting A0 = 13,000, we have:
13,000 = k · 13,000 - 800

Simplifying the equation, we get:
800 = k · 13,000

Dividing both sides by 13,000, we get:
k = 800 / 13,000

Simplifying the fraction, we get:
k = 0.0615

Therefore, k = 0.0615.

(b) To find the balance after seven months, we can use the recursive formula again.
A1 = k · A0 - 800
A2 = k · A1 - 800
A3 = k · A2 - 800
A4 = k · A3 - 800
A5 = k · A4 - 800
A6 = k · A5 - 800
A7 = k · A6 - 800

Substituting the values of A0 and k, we have:
A0 = 13,000
K = 0.0615

A1 = 0.0615 · 13,000 - 800
A2 = 0.0615 · (0.0615 · 13,000 - 800) - 800
A3 = 0.0615 · (0.0615 · (0.0615 · 13,000 - 800) - 800) - 800
A4 = 0.0615 · (0.0615 · (0.0615 · (0.0615 · 13,000 - 800) - 800) - 800) - 800
A5 = 0.0615 · (0.0615 · (0.0615 · (0.0615 · (0.0615 · 13,000 - 800) - 800) - 800) - 800) - 800
A6 = 0.0615 · (0.0615 · (0.0615 · (0.0615 · (0.0615 · (0.0615 · 13,000 - 800) - 800) - 800) - 800) - 800) - 800
A7 = 0.0615 · (0.0615 · (0.0615 · (0.0615 · (0.0615 · (0.0615 · (0.0615 · 13,000 - 800) - 800) - 800) - 800) - 800) - 800) - 800

Evaluating this expression, we find:
A7 ≈ $7,792.99

Therefore, her balance after seven months is approximately $7,792.99.