Answer
See explanation
Work Step by Step
**Solution Explanation**
We have the sequence defined by
\[
t_n = 2 + n, \quad \text{for all integers } n \ge 0.
\]
We want to show that it satisfies the recurrence
\[
t_k = 2\,t_{k-1} \;-\; t_{k-2},
\quad \text{for all integers } k \ge 2.
\]
---
### Step 1. Write \(t_k\), \(t_{k-1}\), and \(t_{k-2}\) from the Formula
From \(t_n = 2 + n\):
\(t_k = 2 + k.\)
\(t_{k-1} = 2 + (k - 1) = k + 1.\)
\(t_{k-2} = 2 + (k - 2) = k.\)
---
### Step 2. Verify the Recurrence
Compute \(2\,t_{k-1} - t_{k-2}\):
\[
2 \, t_{k-1} - t_{k-2}
\;=\;
2(k + 1) \;-\; k
\;=\;
2k + 2 - k
\;=\;
k + 2
\;=\;
2 + k
\;=\;
t_k.
\]
Hence,
\[
t_k = 2\,t_{k-1} - t_{k-2},
\]
as required.
