If n be integer gt1, then prove that `sum_(r=1)^(n-1) cos (2rpi)/n=-1`

Evaluate sum_(r=1)^(n-1) cos^(2) ((rpi)/(n)) .

Prove that sum_(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)^n C_r=1/n .

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

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

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. The value of sum_(r=0)^(n-1) a_(r) is

If x+y=1, prove that sum_(r=0)^n .^nC_r x^r y^(n-r) = 1 .

Prove that ((n + 1)/(2))^(n) gt n!

If n is a positive integer, prove that underset(r=1)overset(n)sumr^(3)((""^(n)C_(r))/(""^(n)C_(r-1)))^(2)=((n)(n+1)^(2)(n+2))/(12)

Prove that sum_(r=0)^ssum_(s=1)^n^n C_s^n C_r=3^n-1.

The value of [100(x-1)] is where [x] is the greatest integer less than or equal to x and x=(sum_(n=1)^44 cos n^@)/(sum_(n=1)^44 sin n^@)