If ` (9 + 4 sqrt(5))^(n) = I + f `, `n` and `I` being positive integers and `f` is a proper fraction, show that ` (I-1 ) f + f^(2)` is an even integer.

Explanation

`(9 + 4sqrt(5))^(n) = I + f ` …(i) ,brgt ` 0 le f lt 1 ` …(ii)
Let ` f' = (9- 4 sqrt(5))^(n)` …(iii)
and ` 0 lt f' ,t 1 ` …(iv)
From Eqs .(i) and (iii), we get
` I + f + f' = (9 + 4 sqrt(5))^(n) + (9 - 4 sqrt(5))^(n)`
`= 2{9^(n) + ""^(n)C_(2) 9^(n-2)(4sqrt(5))^(2) + ...}`
2N , , where N is a positive integer .
and from Eqs.(ii) and (iii), we get ` 0 lt f + f' lt 2`
Since , f + f' is an integer.
` therefore f + f' = 1`
Now , ` I + 1 = 2N rArr 1 = 2N - 1` ...(v)
`because (I +f) (1 -f) = (9 + 4sqrt(5))^(n) f'`
`= (9 + 4sqrt(5))^(n) (9-4sqrt(5))^(n) = 1^(n) = 1`
` therefore (I - 1) f+ f^(2) = I - 1 = 1N - 1 - 1 =2N = -2`
[from Eq.(v)]
= An even integer .