To solve this problem, we need to determine the appropriate recursive formula for calculating the son's weekly allowance in future years. The problem specifies that the weekly allowance starts at \$10 and changes each subsequent year by doubling the previous year's allowance and then subtracting 8.
Let's analyze each given option to find the correct recursive formula:
1. Option (1):
[tex]\[
a_n = 2n - 8
\][/tex]
This formula is not recursive. Instead, it's an explicit formula (although not correctly suited for this context). It expresses [tex]\(a_n\)[/tex] directly in terms of [tex]\(n\)[/tex], which does not match the problem’s pattern of calculating each year based solely on the previous year's amount.
2. Option (2):
[tex]\[
a_n = 2(n + 1) - 8
\][/tex]
This is another explicit formula, similar to option (1). It does not depend on the previous year's allowance [tex]\(a_{n-1}\)[/tex] and therefore does not match the recursive nature needed.
3. Option (3):
[tex]\[
\begin{cases}
a_1 = 10 \\
a_{n+1} = 2a_n - 8
\end{cases}
\][/tex]
This recursive formula starts by setting [tex]\(a_1 = 10\)[/tex]. For each subsequent year, it calculates the allowance by doubling the previous year's allowance ([tex]\(2a_n\)[/tex]) and then subtracting 8. This matches perfectly with the given allowance recalculation rule.
4. Option (4):
[tex]\[
\begin{cases}
a_1 = 10 \\
a_{n+1} = 2(a_n - 8)
\end{cases}
\][/tex]
Though this formula also involves recursion, it is structured incorrectly. It first subtracts 8 from the previous year's allowance and then doubles it, which does not align with the described recalculation rule of doubling first and then subtracting 8.
Based on our examination, the correct recursive formula is given by:
[tex]\[ \begin{cases}
a_1 = 10 \\
a_{n+1} = 2a_n - 8
\end{cases} \][/tex]
Therefore, the correct answer is the formula described in option (3).
