If the value of `prod_(k=1)^(50)[{:(1,2k-1),(0,1):}]` is equal to `[{:(1,r),(0,1):}]` then `r` is equal to

The value of prod_(r=1)^(7)cos\ (pi r)/(15) is

if the product of n matrices [(1,1),(0,1)][(1,2),(0,1)][(1,3),(0,1)]…[(1,n),(0,1)] is equal to the matrix [(1,378),(0,1)] the value of n is equal to

The value of sum_(r=0)^(20)(-1)^(r )(""^(50)C_(r))/(r+2) is equal to

The value of sum_(r=1)^(n) (-1)^(r+1)(""^(n)C_(r))/(r+1) is equal to

The value of sum_(r=0)^(2n)(-1)^(r)*(""^(2n)C_(r))^(2) is equal to :

sum _(k=1)^(n) tan^(-1). 1/(1+k+k^(2)) is equal to

A=[{:(1,0,k),(2, 1,3),(k,0,1):}] is invertible for

If the value of the integral I=int_(0)^(1)(dx)/(x+sqrt(1-x^(2))) is equal to (pi)/(k) , then the value of k is equal to

The value of lim_(xrarr1)Sigma_(r=1)^(10)=(x^(r )-1^(r ))/(2(x-1)) is equal to

The value of 5 * cot ( Sigma_(k =1)^(5) cot ^(-1) ( k^(2) + k + 1)) is equal to