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 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 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 (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.

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 n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is (1) an irrational number (2) an odd positive integer (3) an even positive integer (4) a rational number other than positive integers

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 (9+4sqrt(5))^(n)=p+beta , where n and p are positive integers and beta is a positive proper fraction, prrove that (1-beta)(p+beta)=1 and p is an odd integer.

If n is a positive integer, prove that |I m(z^n)|lt=n|I m(z)||z|^(n-1.)

If in a positive integer such that If a number a=p+f whre p is an integer and 0ltflt1 . Here p is called the integral part of a and f its fractional part. Let n in N and (sqrt(3)+1)^(2n)=p+f , where p is the integral part and 0ltflt1 . On the basis of bove informationi answer teh following question: The integral part p of (sqrt(3)+1)^(2n) is (A) an even number for al n epsilon N (B) an odd number for all nepsilon N (C) anodd or even number according as n is odd or even (D) an even or odd nuber according as n is odd or even