Show that the integral part of
` (5 sqrt(5) + 11)^(2n +1)` is even where ` n in N ` .

Explanation

`(5 sqrt(5) + 11)^(2n +1) `can ve written as `(sqrt(125) + 11)^(2n +1)`
Now , let ` I + f = (sqrt(125) + 11)^(2n+1)` …(i)
and let ` f'= (sqrt(125) - 11)^(2n+1)` …(iii)
` 0 lt f' lt 1` …(iv)
On subtracting Eq. (iii) from (i) , we get
` I + f - f' (sqrt(125) + 11)^(2n + 1) - (sqrt(125) - 11)^(2n + 1)`
` I + 0 = 2p , AA p in ` N = Even integer [from theorem 1]
` I = 2p ` = Even integer .