Answer
See explanation
Work Step by Step
**Solution Explanation**
We have a sequence defined by the formula
\[
c_n = 2^n \;-\; 1, \quad \text{for all integers } n \ge 0.
\]
We want to show that it satisfies the recurrence
\[
c_k = 2\,c_{k-1} \;+\; 1, \quad \text{for all integers } k \ge 1.
\]
---
### Step 1. Express \(c_k\) and \(c_{k-1}\) From the Given Formula
\(c_k = 2^k - 1.\)
\(c_{k-1} = 2^{k-1} - 1.\)
---
### Step 2. Verify the Recurrence
Compute the right-hand side of the recurrence:
\[
2 \, c_{k-1} \;+\; 1
\;=\;
2 \bigl(2^{k-1} - 1\bigr) \;+\; 1
\;=\;
2^k - 2 + 1
\;=\;
2^k - 1
\;=\;
c_k.
\]
Hence,
\[
c_k = 2 \, c_{k-1} + 1 \quad \text{for all } k \ge 1,
\]
as required.
