Chapter 12.5, Problem 7E

To determine

To Prove: $1\cdot 2+2\cdot 3+3\cdot 4+\mathrm{......}+n\left(n+1\right)=\frac{n\left(n+1\right)\left(n+2\right)}{3}$ is True for all natural numbers $n$ .

Explanation of Solution

Given information:

Use Mathematical induction to prove the above mentioned formula.

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

  $1\cdot 2+2\cdot 3+3\cdot 4+\mathrm{......}+n\left(n+1\right)=\frac{n\left(n+1\right)\left(n+2\right)}{3}$

Put $n=1$

We get

Left hand side = $1\cdot 2=2$

Right hand side = $\frac{1\left(1+1\right)\left(1+2\right)}{3}=\frac{1\left(2\right)\left(3\right)}{3}=2$

So Left hand side = Right hand side

It True for $n=1$ .

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

So

  $1\cdot 2+2\cdot 3+3\cdot 4+\mathrm{......}+k\left(k+1\right)=\frac{k\left(k+1\right)\left(k+2\right)}{3}$ is True

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

Left hand side = $1\cdot 2+2\cdot 3+3\cdot 4+\mathrm{......}+k\left(k+1\right)+\left(k+1\right)\left(k+1+1\right)$

  $\Rightarrow$ Left hand side = $\frac{k\left(k+1\right)\left(k+2\right)}{3}+\left(k+1\right)\left(k+2\right)$

Taking $\left(k+1\right)\left(k+2\right)$ common from both terms

We get

  $\Rightarrow$ Left hand side = $\left(k+1\right)\left(k+2\right)\left(\frac{k}{3}+1\right)$

  $\Rightarrow$ Left hand side = $\left(k+1\right)\left(k+2\right)\left(\frac{k+3}{3}\right)$

  $\Rightarrow$ Left hand side = $\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)}{3}$

Right hand side = $\frac{\left(k+1\right)\left(k+1+1\right)\left(k+1+2\right)}{3}$

  $\Rightarrow$ Right hand side = $\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)}{3}$

Left hand side = Right hand side

And hence the proof by Mathematical induction.