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Question Number 62169 by Tawa1 last updated on 16/Jun/19

$$\mathrm{Prove}\mathrm{without}\mathrm{induction}\mathrm{that}:{(1+\sqrt{2})}^{2\mathrm{n}}+{(1-\sqrt{2})}^{2\mathrm{n}}\mathrm{is}\mathrm{even}\mathrm{for}\mathrm{every}$$$$\mathrm{natural}\mathrm{number}\mathrm{n}.$$

Answered by ajfour last updated on 16/Jun/19

$${(1+\sqrt{2})}^{2n}+{(1-\sqrt{2})}^{2n}$$$$=2\underset{r=0}{\sum ^n}{}^{2n}{C}_{2r}{\left(\sqrt{2}\right)}^{2r}\left(oddtermscancel\right)$$$$=2\left(integer\right)2^r=even.$$

Commented by mr W last updated on 16/Jun/19

$$will{(1+\sqrt{2})}^n+{(1-\sqrt{2})}^{n}alsodo?$$

Commented by Tawa1 last updated on 16/Jun/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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