Prove that `underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0` , where `k = 3n//2` and n is an even integer.

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

sum_(r=1)^k(−3)^(r−1) .^(3n)C_(2r−1)=0, where k=(3n)/2 and n is an even integer

underset(r=1)overset(n)(sum)r(.^(n)C_(r)-.^(n)C_(r-1)) is equal to

Prove that overset(n)underset(r=0)(Sigma^(n))C_(r).4^(r)=5^(n)

Prove that overset(n)underset(r=0)(Sigma^(n))C_(r).4^(r)=5^(n)

Statement-1: If underset(r=1)overset(n)(sum)r^(3)((.^(n)C_(r))/(.^(n)C_(r-1)))^(2)=196 , then the sum of the coeficients of powerr of xin the expansion of the polynomial (x-3x^(2)+x^(3))^(n) is -1. Statement-2: (.^(n)C_(r))/(.^(n)C_(r-1))=(n-r+1)/(r) AA n in N and r in W .

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

If underset(k=0)overset(n)(sum)(k^(2)+k+1)k! =(2007).2007! , then value of n is

Prove that (3!)/(2(n+3))=sum_(r=0)^n(-1)^r((^n C_r)/(^(r+3)C_r))

The sum S_(n)=sum_(k=0)^(n)(-1)^(k)*^(3n)C_(k) , where n=1,2,…. is