Consider the following statements I.` n (sin^2 (67 1/2^@) -sin^2 (22 1/2^@)) gt 1` for all positive integers `n geq 2`.II. If `x` is any positive real number, then `nx gt 1` for all positive integers `n geq 2`.Which of the above statement(s) is/are correct?

Consider the following statements: I. n(sin^2 67(1^@)/(2)-sin^2 22 (1^@)/(2)) gt 1 for all positive integers n ge 2 . II. If x is any positive real number, then nx gt 1 for all positive integers n ge 2 . Which of the above statements is/are correct?

Prove that 2^n gt n for all positive integers n.

Prove that 2^(n) gt n for all positive integers n.

Prove that 2^(n) gt n for all positive integers n.

Prove that 2^(n) gt n for all positive integers n.

1/(n+1) + 1/(n+2)+…..+ 1/(3n+1) gt 1 for every positive integer n.

Consider the following statements: 1. a^n+b^n is divisible by a+b if n=2k+1, where k is a positive integer. 2. a^n-b^n is divisible by a-b if n=2k , where k is a positive integer. Which of the statements given above is /are correct? (a) 1 only (b) 2 only (c) both 1 and 2 (d) Neither 1 nor 2

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4

If x and y are positive real numbers and m, n are any positive integers, then Prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt =1/4