Select the correct answer.
Aaron borrows $150 from his friend Austin. He promises to pay back the money in 4 monthly installments. Each month he wants to pay half the amount he paid the previous month. Assuming Austin does not charge any interest, how much should Aaron pay the first month to repay the money as scheduled?
A.
$60
B.
$70
C.
$80
D.
$90
E.
$100
1 answer
Each month, Aaron will pay half of what he paid the previous month. This is a geometric sequence with first term $x$ (the first payment), common ratio $\frac{1}{2}$ (since each payment is half of the previous one), and four terms. Therefore, the sum of the sequence is given by: $$S = \frac{x\left(1-\frac{1}{2^4}\right)}{1-\frac{1}{2}} = 2x\left(1-\frac{1}{16}\right) = \frac{31}{16}x$$ Since $S$ is equal to the amount Aaron borrowed ($150), we have $\frac{31}{16}x = 150$. Solving for $x$, we get $x=\frac{480}{31} \approx \boxed{\textbf{(B)}\ $70}$.
1 answer