If m and n are positive quantites , prove that `((mn+1)/(m+1))^(m+1)ge n^(m)`

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

If m and n are positive integers, then prove that the coefficients of x^(m) " and " x^(n) are equal in the expansion of (1+x)^(m+n)

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

If m is negative or positive and greater than 1 , show that 1^(m)+3^(m)+5^(m)+….+(2n-1)^(m) gt n^(m+1)

If n ge 1 is a positive integer, then prove that 3^(n) ge 2^(n) + n . 6^((n - 1)/(2))

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

If a ,\ m ,\ n are positive integers, then {root(m)root (n)a}^(m n) is equal to (a) a^(m n) (b) a (c) a^(m/n) (d) 1

If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p terms, then prove that (m+n)(1/m-1/p)=(m+p)(1/m-1/n)dot

If k a n d n are positive integers and s_k=1^k+2^k+3^k++n^k , then prove that sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+1)dot