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Q31E

Expert-verifiedFound in: Page 330

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

Prove that 2 divides ${n}^{2}+n$ whenever n is a positive integer.

2 divides ${n}^{2}+n$ whenever n is a positive integer

If n=1,

${1}^{2}+1=2$

it is true for n=1.

Let P(k) be true.

${k}^{2}+k$

We need to prove that P(k+1) is true.

$\begin{array}{r}(k+1{)}^{2}+(k+1)\\ ={k}^{2}+3k+2\\ =\left({k}^{2}+k\right)+(2k+2)\\ =\left({k}^{2}+k\right)+2(k+1)\end{array}$

It is true for P(k+1) is true.

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