Worked Example 4

For each of the following recurrence relations, explain how to obtain the next term in the sequence based on the given formula, describe the relationship represented by the recurrence relation, and calculate and write down the first \(5\) terms of the sequence starting from the initial value provided.

1. \({{x}_{n+1}}=\sqrt{{{x}_{n}}+5}\), \({{x}_{0}}=1\).
2. \({{u}_{n+1}}=0.5{{u}_{n}}+8\), \({{u}_{1}}=4\).
3. \({{a}_{n}}={{a}_{n-1}}+k\), \({{a}_{0}}=3\).
4. \({{u}_{n}}=-2{{u}_{n-1}}\), \({{u}_{0}}=3\).
5. \({{T}_{n}}={{T}_{n-1}}+{{T}_{n-2}}\), \({{T}_{1}}=0\), \({{T}_{2}}=1\).