To determine how Pepe's and Leo's balances will compare after a very long time, let's first analyze the information given in the table and identify the patterns in their savings.
From the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Month} & \text{Leo} & \text{Pepe} \\
\hline
1 & \$50 & \$2 \\
\hline
2 & \$70 & \$4 \\
\hline
3 & \$90 & \$8 \\
\hline
4 & \$110 & \$16 \\
\hline
5 & \$130 & \$32 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Analysis:
1. Identifying Leo's Pattern:
Leo's balance increases by a fixed amount each month. Let's denote his balance in the [tex]\(n\)[/tex]-th month as [tex]\(B_{\text{Leo}_n}\)[/tex].
- Month 1: [tex]\( \$50 \)[/tex]
- Month 2: [tex]\( \$70 \)[/tex]
- Month 3: [tex]\( \$90 \)[/tex]
- Month 4: [tex]\( \$110 \)[/tex]
- Month 5: [tex]\( \$130 \)[/tex]
His balance increases by \[tex]$20 each month. This indicates an arithmetic sequence where the balance can be expressed as:
\[
B_{\text{Leo}_n} = 50 + 20(n - 1)
\]
2. Identifying Pepe's Pattern:
Pepe's balance doubles each month. Let's denote his balance in the \(n\)-th month as \(B_{\text{Pepe}_n}\).
- Month 1: \( \$[/tex]2 \)
- Month 2: [tex]\( \$4 \)[/tex]
- Month 3: [tex]\( \$8 \)[/tex]
- Month 4: [tex]\( \$16 \)[/tex]
- Month 5: [tex]\( \$32 \)[/tex]
This is a geometric sequence where each term is twice the previous term. His balance can be expressed as:
[tex]\[
B_{\text{Pepe}_n} = 2 \cdot 2^{(n - 1)}
\][/tex]
3. Comparing the Balances Over Time:
- For Leo, the balance increases linearly with a constant addition of \[tex]$20 each month.
- For Pepe, the balance increases exponentially as it doubles every month.
Even though Leo starts with a higher balance (\$[/tex]50) compared to Pepe (\$2), the exponential growth of Pepe's balance means that eventually, his balance will surpass Leo's balance.
4. Long-Term Behavior:
- Since exponential growth outpaces linear growth, after a sufficient number of months, Pepe's balance will become greater than Leo's balance. This is evident as [tex]\(2^{(n-1)}\)[/tex] grows much faster than the linear increment of 20 per month.
### Conclusion:
After a very long time, Pepe's balance will be greater than Leo's balance.
Therefore, the correct answer is:
A. Pepe's balance will be greater.
