Answer
\[
a_{25} = 7 \times 16{,}777{,}216 = 117{,}440{,}512.
\]
Work Step by Step
We have the following information:
- For any integer \(k > 1\), the number of operations \(a_k\) for input size \(k\) is twice the number of operations \(a_{k-1}\) for input size \(k-1\).
- For input size \(1\), the algorithm executes \(a_1 = 7\) operations.
Formally, the recurrence is
\[
a_k = 2\,a_{k-1} \quad \text{for } k \ge 2,
\quad\text{with } a_1 = 7.
\]
This is a geometric sequence with common ratio 2. The closed‐form solution is:
\[
a_k = a_1 \cdot 2^{\,k-1} = 7 \cdot 2^{\,k-1}.
\]
Hence, for \(k = 25\):
\[
a_{25} = 7 \cdot 2^{24}.
\]
You can leave the answer in this exact form, or if needed, you can compute the numerical value:
\[
2^{10} = 1024,\quad 2^{20} = (2^{10})^2 = 1024^2 = 1{,}048{,}576, \quad 2^{24} = 2^{20} \times 2^4 = 1{,}048{,}576 \times 16 = 16{,}777{,}216.
\]
