Chapter 10.7, Problem 21PPS

To determine

To find a counterexample to disprove that

  $1+2+3+\mathrm{......}+n=n^2$ .

Answer to Problem 21PPS

For $n=2$ .

Explanation of Solution

Given:

  $1+2+3+\mathrm{......}+n=n^2$

Calculation:

In order to find a counterexample to disprove that $1+2+3+\mathrm{......}+n=n^2$

First, let’s check this is true for $n=1$ .

For $n=1$ , left side:

  $1=1$

Right side:

  $\begin{array}{c}{1}^2=1\\ =\text{Left side}\end{array}$

Thus, the given identity is true for $n=1$ .

For $n=2$ , left side:

  $1+2=3$

Right side:

  $\begin{array}{c}{2}^2=4\\ \ne \text{Left side}\end{array}$

Thus, the given identity is not true for $n=2$ .

Thus, counter example to disprove the given statement is for $n=2$ .