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_(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_(2)s_(1)+^(n+1)C_(3)s_(2)+......+^(n+1)C_(n+1)s_(n)=2^(n)S_(n)

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 ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot

Prove that ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot

Let S_n=1+q+q^2 +...+q^n and T_n =1+((q+1)/2)+((q+1)/2)^2+...((q+1)/2)^n If alpha T_100=^101C_1 +^101C_2 x S_1 ...+^101C_101 x S_100, then the value of alpha is

Let S_n=1+q+q^2 +...+q^n and T_n =1+((q+1)/2)+((q+1)/2)^2+...((q+1)/2)^n If alpha T_100=^101C_1 +^101C_2 x S_1 ...+^101C_101 x S_100, then the value of alpha is equal to (A) 2^99 (B) 2^101 (C) 2^100 (D) -2^100

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

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

if S_n = C_0C_1 + C_1C_2 +...+ C_(n-1)C_n and S_(n+1)/S_n = 15/4 then n is