To determine how much Aaron should pay in the first month, let's break down the problem step-by-step:
1. Define the total amount borrowed: Aaron borrows [tex]$150 from his friend.
2. Represent the payments Aaron needs to make:
- Let \( x \) be the amount Aaron pays in the first month.
- According to the problem, he pays half the amount he paid in the previous month for each subsequent month.
Therefore:
- First month's payment: \( x \)
- Second month's payment: \( \frac{x}{2} \)
- Third month's payment: \( \frac{x}{4} \)
- Fourth month's payment: \( \frac{x}{8} \)
3. Set up an equation for the total amount: Aaron promises to repay the total amount of $[/tex]150 in these 4 installments. So, we sum up the payments and set them equal to [tex]$150:
\[
x + \frac{x}{2} + \frac{x}{4} + \frac{x}{8} = 150
\]
4. Sum the series: This expression is a geometric series where each term is half of the previous one. The series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \) can be summed up as follows:
\[
1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8}
\]
5. Express the total payment in terms of \( x \):
\[
x \left( \frac{15}{8} \right) = 150
\]
6. Solve for \( x \):
\[
x = 150 \times \frac{8}{15}
\]
\[
x = 80
\]
Therefore, Aaron should pay \$[/tex]80 in the first month to repay the money as scheduled.
Thus, the correct answer is:
C. \$80
