If R, n are positive integers, n is odd, `0lt F lt1" and if "`
`(5sqrt(5)+11)^(n)=R+F,` then prove that
`(R+F).F=4^(n)`

If R, n are positive integers, n is odd, 0lt F lt1" and if "(5sqrt(5)+11)^(n)=R+F, then prove that R is an even integer and

If I, n are positive integers , 0 lt f lt 1 and if (7 + 4sqrt3)^n = I +f , then show that I is an odd integer

If I, n are positive integers , 0 lt f lt 1 and if (7 + 4sqrt3)^n = I +f , then show that (I+f) (1 - f) = 1

If n is a positive integer and (3sqrt(3)+5)^(2n+1)=l+f where l is an integer annd 0 lt f lt 1 , then

If n gt 0 is an odd integer and x = (sqrt(2) + 1)^(n), f = x - [x], " then" (1 - f^(2))/(f) is

If (2+sqrt(3))^n=I+f, where I and n are positive integers and 0lt f lt1 , show that I is an odd integer and (1-f)(1+f) =1

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.

Let (6sqrt(6)+14)^(2n+1)=R , if f be the fractional part of R, then prove that Rf=20^(2n+1)

If f(x) = n , where n is an integer such that n le x lt n +1 , the range of f(x) is

If N denotes the set of all positive integers and if f : N -> N is defined by f(n)= the sum of positive divisors of n then f(2^k*3) , where k is a positive integer is