수열의 귀납적 정의

Let $P(n)$ be, “$L_n = 2f_{n+1} - f_n$.” First, verify that $P(1)$ and $P(2)$ are true. Now let $k$ be a natural number with $k \geq 2$ and assume that $P(1)$, $P(2)$, …, $P(k)$ are all true. Since $P(k)$ and $P(k-1)$ are both assumed to be true, we can use them to help prove that $P(k+1)$ must then be true as follows:

$$\begin{aligned}L_{k+1} & = L_k + L_{k-1} \\ & = \left( 2f_{k+1} - f_k \right) + \left( 2f_k - f_{k-1} \right) \\ & = 2 \left( f_{k+1} + f_k \right) - \left( f_k + f_{k-1} \right) \\ & = 2f_{k+2} - f_{k+1}\end{aligned}$$

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