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

Question Number 58409 by Tawa1 last updated on 22/Apr/19

Prove without mathematical induction that the

expression {(1 + \sqrt{2})}^{2 n} + {(1 - \sqrt{2})}^{2 n} is even for every

natural number n .

Commented by maxmathsup by imad last updated on 22/Apr/19

l e t A_{n} = {(1 + \sqrt{2})}^{2 n} + {(1 - \sqrt{2})}^{2 n} \Rightarrow A_{n} = \sum_{k = 0}^{2 n} C_{2 n}^{k} {(\sqrt{2})}^{k} + \sum_{k = 0}^{2 n} C_{2 n}^{k} {(- \sqrt{2})}^{k}

= \sum_{k = 0}^{2 n} C_{2 n}^{k} {{(\sqrt{2})}^{k} + {(- \sqrt{2})}^{k}} = \sum_{k = 2 p} (\dots) + \sum_{k = 2 p + 1} (\dots)

= \sum_{p = 0}^{n} C_{2 n}^{2 p} 2^{p + 1} + 0 = 2 {\sum_{p = 0}^{n} C_{2 n}^{2 p} 2^{p}} \Rightarrow A_{n} i s e v e n .

Commented by Tawa1 last updated on 23/Apr/19

God bless you sir, i will check if i can understand the solution

Commented by maxmathsup by imad last updated on 23/Apr/19

y o u a r e w e l c o m e s i r, h e r e i h a v e u s e d t h e b i n o m e f o r m u l a e .