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 $n\geq 3$.

Solution

Verified$\textbf{Fibonacci sequence}$: 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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