Question

Hey! I need help with this question. I know the answer, I need to understand how to get to that answer (with details and explanation)

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

Answer 1

Or using geometric sequence:

`S_n=(a_1(1-r^n))/(1-r)`

`S_4=150\\r=(1)/(2)\\n=4\\a_1=?`

`150=(a_1\left(1-\left((1)/(2)\right)^4\right))/(1-(1)/(2))\\\\150=(a_1\left(1-(1)/(16)\right))/((1)/(2))\\\\75=a_1\cdot(15)/(16)\\\\a_1=80`

Answer 2

To solve the problem, we can work backwards from the final payment to the first payment.

Let X be the first payment Aaron makes. Then, his second payment is X/2, his third payment is (X/2)/2 = X/4, and his fourth payment is (X/4)/2 = X/8. The sum of these payments must be equal to $150:

X + X/2 + X/4 + X/8 = 150

We can simplify this equation by multiplying both sides by 8 to eliminate the fractions:

8X + 4X + 2X + X = 1200

15X = 1200

X = 80

Therefore, the first payment Aaron should make is $80, which is option C.