Chapter 12.5, Problem 3E

To determine

To Prove: $2+4+6+8+\mathrm{........}+2n=n\left(n+1\right)$ 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

  $2+4+6+8+\mathrm{........}+2n=n\left(n+1\right)$

Put $n=1$

We get

Left hand side = 2

Right hand side = $n\left(n+1\right)=1\left(1+1\right)=2$

So Left hand side = Right hand side

It True for $n=1$ .

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

So

  $2+4+6+8+\mathrm{........}+2k=k\left(k+1\right)$ is True

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

Left hand side = $2+4+6+8+\mathrm{........}+2\left(k+1\right)$

  $\Rightarrow$ Left hand side = $2+4+6+8+\mathrm{.......}+2k+2\left(k+1\right)$

  $\Rightarrow$ Left hand side = $k\left(k+1\right)+2\left(k+1\right)$

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

We get

  $\Rightarrow$ Left hand side = $\left(k+1\right)\left(k+2\right)$

Right hand side = $\left(k+1\right)\left(k+1+1\right)$

  $\Rightarrow$ Right hand side = $\left(k+1\right)\left(k+2\right)$

We got

Left hand side = Right hand side

And hence the proof by Mathematical induction.