Chapter 12.5, Problem 31E

To determine

To Prove: $F_1{}^2+F_2{}^2+F_3{}^2+\mathrm{.......}+F_n{}^2=F_n\cdot F_{n+1}$

Explanation of Solution

Given information:

Use Mathematical induction to prove the above mentioned formula.

  $\begin{array}{l}{F}_1=1\\ F_2=1\\ F_n=F_{n-1}+F_{n-2}\end{array}$ .

Concept used:

1.By using Mathematical induction technique we need to prove that this formula holds True for all natural numbers. The first step is to prove True for $n=1$ .

2.Second step by using induction hypothesis of mathematical induction we assume for $n=k$ given formula is True.

3.Third step is we need to prove that above formula is True for $n=k+1$ by using formula is True for $n=k$ .

Proof:

First, we need to prove the above formula is True for $n=1$ .

Given formula is

  $F_1{}^2+F_2{}^2+F_3{}^2+\mathrm{.......}+F_n{}^2=F_n\cdot F_{n+1}$

Put $n=1$

We get

Left hand side = $F_1{}^2=1^2=1$

Right hand side = $F_1\cdot F_{1+1}=F_1\cdot F_2=1\cdot 1=1$

So

Left hand side = Right hand side

It True for $n=1$ .

Let us assume it is True for $n=k$

So

  $F_1{}^2+F_2{}^2+F_3{}^2+\mathrm{.......}+F_k{}^2=F_k\cdot F_{k+1}$ is True.

Therefore

We need to prove it is True for $n=k+1$

Left hand side = $F_1{}^2+F_2{}^2+F_3{}^2+\mathrm{.......}+F_k{}^2+F_{k+1}{}^2$

  $\Rightarrow$ Left hand side = $F_{k+1}{}^2+F_k\cdot F_{k+1}$

  $\Rightarrow$ Left hand side = $\left(F_{k+1}+F_k\right)\cdot F_{k+1}$

  $\Rightarrow$ Left hand side = $F_{k+1}\cdot F_{k+2}$

Right hand side = $F_{k+1}\cdot F_{k+1+1}$

  $\Rightarrow$ Right hand side = $F_{k+1}\cdot F_{k+2}$

Left hand side =Right hand side

Therefore

  $F_1{}^2+F_2{}^2+F_3{}^2+\mathrm{.......}+F_n{}^2=F_n\cdot F_{n+1}$ is True for $n=k+1$ .

And hence the proof by Mathematical induction.