Chapter EP, Problem 11.7.1EP

To determine

State the proof of the given statement.

Explanation of Solution

Given:

The given statement is $2+4+6+\cdot \cdot \cdot +2n=n^2+n$ for all natural numbers.

Calculation:

Use mathematical induction.

For $n=1$ .

L.H.S $=2+4+6+\cdot \cdot \cdot +2n=2n=2\cdot 1=2$

R.H.S $=n^2+n=1^2+1=2$

L.H.S $=$ R.H.S

The statement is true for $1$ .

For $n=m$

  $2+4+6+\cdot \cdot \cdot +2m=m\left(m+1\right)$

For $n=m+1$

  $\begin{array}{l}=2+4+6+\cdot \cdot \cdot +2m+2\left(m+1\right)\\ \left[2+4+6+\cdot \cdot \cdot +2m=m\left(m+1\right)\right]\\ =m\left(m+1\right)+2\left(m+1\right)\\ =\left(m+1\right)\left(m+2\right)\\ =\left(m+1\right)\left\{\left(m+1\right)+1\right\}\end{array}$

The given statement is true for $n=m+1$ .

Hence the given statement is true for all natural numbers.