(5.4.5) Inductive Proof Technique for a Recursive Sequence Formula

This presentation details the process of proving the formula e_n = 5 * 3^n + 7 * 2^n for a sequence defined by e_0 = 12, e_1 = 29, and for k ≥ 2, e_k = 5e_(k-1) - 6e_(k-2). The proof employs mathematical induction, starting with verifying the formula for base cases n = 0 and n = 1. The inductive step involves assuming the formula is true for all integers up to k, and then proving it for k + 1. By substituting the inductive assumption into the recursive definition, the formula for e_(k+1) is shown to match the pattern of the hypothesis, thus proving the formula holds for every integer n ≥ 0. The procedure exemplifies the application of inductive reasoning in confirming the validity of expressions for sequences defined recursively.

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