Define a sequence $\{g_n\}$ as $g_1=c_1$ and $g_2=c_2$ for constants $c_1$ and $c_2$ , and $g_n=g_{n-1}+f_{n-2}$ for $n\geq 3$ . Prove that $g_n=g_1f_{n-2}+g_2f_{n-1}$ for all $n\geq 3$ .

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$\textbf{Fibonacci sequence}$ : the list of integers $f_n$ where $f_1=1$ , $f_2=1$ , $f_n=f_{n-1}+f_{n-2}$ for $n\geq 3$ .

Given:

$g_1=c_1$

g_2=c_2

$g_n=g_{n-1}+g_{n-2}$ for $n\geq 3$

To proof: $g_n=g_1f_{n-2}+g_2f_{n-1}$ for all $n\geq 3$ .

$\textbf{PROOF BY STRONG INDUCTION}$

Let $P(n)$ be the statement " $g_n=g_1f_{n-2}+g_2f_{n-1}$ ".

$\textbf{Basis step}$ $n=3$ and $n=4$

\begin{align*} g_3&=g_2+g_1=g_1(1)+g_2(1)=g_1f_1+g_2f_2 \\ g_4&=g_3+g_2=g_2+g_1+g_2=g_1(1)+g_2(2)=g_1f_2+g_2f_3 \end{align*}

Thus $P(3)$ and $P(4)$ are true.

$\textbf{Inductive step}$ Let $P(3),P(4),....,P(k)$ be true.

g_i=g_1f_{i-2}+g_2f_{i-1}\text{ for }i=1,2,...,k

We need to proof that $P(k+1)$ is true.

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