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 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 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 (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 in N such that (7 + 4 sqrt(3))^(n) = I + F , then IF is

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 E and F are independent events such that 0 lt P(E) lt 1 and 0 lt P(F) lt 1 , then

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

If n_1, n_2 are positive integers, then (1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i^5)^(n_2) + (1 + i^7)^(n_2) is real if and only if :

If n_1, n_2 are positive integers, then (1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i_5)^(n_2) + (1 + i^7)^(n_2) is real if and only if :