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.
(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.)
$
(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.
(a) To find k, we can substitute the given values into the recursive formula and solve for k.
A0 = 13,000
A1 = k · A0 - 800
Substituting A0 = 13,000:
13,000 = k · 13,000 - 800
Simplifying the equation:
800 = k · 13,000 -13,000
800 = 13,000(k - 1)
Dividing both sides by 13,000:
800 / 13,000 = k - 1
0.0615 = k - 1
Adding 1 to both sides:
0.0615 + 1 = k
k = 1.0615
Therefore, k = 1.0615.
(b) To find her balance after seven months, we can use the recursive formula again:
An = k · An-1 - 800
Substituting n = 7 and k = 1.0615:
A7 = 1.0615 · A6 - 800
Using the recursive formula repeatedly:
A6 = 1.0615 · A5 - 800
A5 = 1.0615 · A4 - 800
A4 = 1.0615 · A3 - 800
A3 = 1.0615 · A2 - 800
A2 = 1.0615 · A1 - 800
A1 = 1.0615 · A0 - 800
A0 = 13,000
Calculating the balance step by step:
A1 = 1.0615 · 13,000 - 800 = 13,755.50
A2 = 1.0615 · 13,755.50 - 800 = 14,518.30
A3 = 1.0615 · 14,518.30 - 800 = 15,315.16
A4 = 1.0615 · 15,315.16 - 800 = 16,148.05
A5 = 1.0615 · 16,148.05 - 800 = 17,019.61
A6 = 1.0615 · 17,019.61 - 800 = 17,932.10
A7 = 1.0615 · 17,932.10 - 800 = 18,888.39
Therefore, her balance after seven months is $18,888.39.