Answer
See explanation
Work Step by Step
**Solution Explanation**
We have a sequence defined by the explicit formula
\[
b_n = 4^n \quad \text{for all integers } n \ge 0.
\]
We want to show that this sequence satisfies the recurrence relation
\[
b_k = 4 \, b_{k-1} \quad \text{for all integers } k \ge 1.
\]
---
### Step 1. Write \(b_k\) and \(b_{k-1}\) from the Given Formula
\(b_k = 4^k.\)
\(b_{k-1} = 4^{\,k-1}.\)
---
### Step 2. Verify the Recurrence
Compute \(4 \, b_{k-1}\):
\[
4 \, b_{k-1} = 4 \cdot 4^{\,k-1} = 4^k = b_k.
\]
Thus, for all \(k \ge 1\),
\[
b_k = 4 \, b_{k-1}.
\]
