Given, `s_n=1+q+q^2+.....+q^n ,S_n=1+(q+1)/2+((q+1)/2)^2+...+((q+1)/2)^n ,q!=1` prove that `"^(n+1)C_1+^(n+1)C_2s_1+^(n+1)C_3s_2+......+^(n+1)C_(n+1)s_n=2^n S_ndot`

Given s=1+q+q^(2)+...+q^(n),S_(n)=1+(q+(1)/(2))+(q+(1)/(2))^(2)+......+(q+(1)/(2))^(n) then prove that ^(n+1)C_(1)+^(n+1)C_(2)s_(1)+......,+^(n+1)C_(n+1)s_(n)=2^(n)s_(n)

Let S_k=1+q+q^2+...+q^k and T_k=1+(q+1)/2+((q+1)/2)^2+...+((q+1)/2)^k q!=1 then prove that sum_(r=1)^(n+1) ^(n+1)C_rS_(r-1)=2^ nT_n

Prove that (^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

If (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a+.^(n)C_(2)a^(2)+ . . +.^(n)C_(n)a^(n) , then prove that .^(n)C_(1)+2.^(n)C_(2)+3.^(n)3C_(3)+ . . .+n.^(n)C_(n)=n.2^(n-1) .

Prove that C_0+(C_1)/(2)+(C_2)/(3)+....+(C_n)/(n+1)=(2^(n+1)-1)/(n+1)

Let S_n=1/1^2 + 1/2^2 + 1/3^2 +….. + 1/n^2 and T_n=2 -1/n , then :

Delta[[ Prove that ,, 11,1,1nC1,(n+1)C1,(n+2)C1(n+1)C2,(n+2)C2,(n+3)C2]]=1

Let S_(n)=1+q+q^(2)+?+q^(n) and T_(n)=1+((q+1)/(2))+((q+1)/(2))^(2)+?+((q+1)/(2)) If alpha T_(100)=^(101)C_(1)+^(101)C_(2)xS_(1)+^(101)C_(101)xS_(100), then the value of alpha is equal to (A) 2^(99)(B)2^(101)(C)2^(100) (D) -2^(100)

If S=sum_(n=1)^(10)(2n+(1)/(2)), then S