Sequences

We are given that and the recursive relation .
We need to find a formula for using mathematical induction.
Step 1: Base Case (Induction Start)
For , we have: This satisfies the base case, so the induction hypothesis holds for .
Step 2: Induction Hypothesis
Assume that the formula holds for some , i.e., assume that:
Step 3: Inductive Step
We now need to prove that the formula holds for , i.e., we need to prove: Using the recursive formula: Substitute the induction hypothesis for : Simplify the expression: Now, expand : Thus, , which completes the inductive step.
Therefore, by the principle of mathematical induction, the formula holds for all , and we have: