[" that,"quad S_(n)=1+q+q^(2)+q^(3)+....+q^(n)],[(q+1)/(2)+((q+1)/(2))^(2)+((q+1)/(2))^(3)+......+((q+1)/(2))^(n)" Then value "],[qquad C_(2)S_(1)+^(n+1)C_(3)S_(2)+......+^(n+1)C_(n+1)S_(n)" is "]

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

{(P+1/(q))^(m).(p-(1)/(q))^(n)}/{(q+(1)/(p))^(m).(q-(1)/(p))^(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

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)

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

(p+iq)^((1)/(n))n+(p-iq)^((1)/(n))=2(p^(2)+q^(2))(1)/(2n)cos((1)/(n)(arctan q)/(p))

Let n inN and k be an integer ge0 such that S_(k)(n)=1^(k)+2^(k)+3^(k)+ . . . +n^(k) Statement-1: S_(4)(n)=(n)/(30)(n+1)(2n+1)(3n^(2)+3n+1) Statement -2: .^(k+1)C_(1)S_(k)(n)+.^(k+1)C_(2)S_(k-1)(n)+ . . . +.^(k+1)C_(k)S_(1)(n)+.^(k+1)C_(k+1)S_(0)(n)=(n+1)^(k+1)-1