Prove-without-induction-that-1-2-2n-1-2-2n-is-even-for-every-natural-number-n-

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

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

natural number n .

Answered by ajfour last updated on 16/Jun/19

{(1 + \sqrt{2})}^{2 n} + {(1 - \sqrt{2})}^{2 n}

= 2 \underset{r = 0}{\sum^{n}}^{2 n} C_{2 r} {(\sqrt{2})}^{2 r} (o d d t e r m s c a n c e l)

= 2 (i n t e g e r) 2^{r} = e v e n .

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

w i l l {(1 + \sqrt{2})}^{n} + {(1 - \sqrt{2})}^{n} a l s o d o ?

Commented by Tawa1 last updated on 16/Jun/19

God bless you sir

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