[" Given "s_(n)=1+q+q^(2)+......+q^(n)&quad s_(n)=1+(q+1)/(2)+((q+1)/(2))^(2)+...+((q+1)/(2))^(n)*q+1],[" prove that "^(n+1)C_(1)+n+1C_(2)*s_(1)+n+1C_(3)*s_(2)+...+n+1-n_(n+1)=2^(n)*S_(n)]

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)

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_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_2s_1+^(n+1)C_3s_2+......+^(n+1)C_(n+1)s_n=2^n S_ndot

If S_(n)=1 + (1)/(2) + (1)/(2) + …..+ (1)/(2^(n-1)), (n in N) then …….

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

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

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)

S_(n) = (1+2+3+....+n)/( n) then S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ..... + S_(n)^(2) =