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Let \(P(n)\) be the statement \(F_0 + F_2 + F_4 + \cdots + F_{2n} = F_{2n+1} - 1\text{.}\) We will show that \(P(n)\) is true for all \(n \ge 0\text{.}\) First the base case is easy because \(F_0 = 0\) and \(F_1 = 1\) so \(F_0 = F_1 - 1\text{.}\) Now consider the inductive case. Assume \(P(k)\) is true, that is, assume \(F_0 + F_2 + F_4 + \cdots + F_{2k} = F_{2k+1} - 1\text{.}\) To establish \(P(k+1)\) we work from left to right:

\begin{align*} F_0 + F_2 + \cdots + F_{2k} + F_{2k+2} ~ \amp = F_{2k+1} - 1 + F_{2k+2} \amp \text{by ind. hyp.}\\ \amp = F_{2k+1} + F_{2k+2} - 1 \amp\\ \amp = F_{2k+3} - 1 \amp \text{by recursive def.} \end{align*}

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