Prove that `sum_(k=1)^(n)(1)/(k(k+1))=1-(1)/(n+1).`

Find sum_(k=1)^(n)(1)/(k(k+1)) .

If A = [((1)/(2),alpha),(0,(1)/(2))] , prove that sum_(k=1)^(n)"det" (A^(k))=(1)/(3)(1-(1)/(4^(n))).

Prove that sum_(k=1)^(n-1)(n-k)cos(2kpi)/n=-n/2 , where ngeq3 is an integer

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Prove that: sum_(k=1)^(100)sin(k x)cos(101-k)x=50"sin"(101 x)

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

Prove that sum_(r=1)^k(-3)^(r-1)^(3n)C_(2r-1)=0,w h e r ek=3n//2 and n is an even integer.

If a_(1),a_(2),a_(3),… are in G.P. , where a_(i) in C (where C satands for set of complex numbers) having r as common ratio such that sum_(k=1)^(n)a_(2k-1)sum_(k=1)^(n)a_(2k+3) ne 0 , then the number of possible values of r is