If `S=1+a+a^2+a^3+a^4+……….to oo` then prove that `a= (S-1)/S`

If S_n=1+r^n+r^(2n)+r^(3n)+.... to oo and s_n=1-r^n+r^(2n)-r^(3n)+... to oo then prove that S_n+s_n=2S_(2n) .

If alpha, beta be the real roots of ax^2+bx+c=0, and s_n=alpha^n + beta^n then prove that as_n + bs_(n-1)+cs_(n-2)=0.for all n in N. Hence or otherwise prove that |(3,1+s_1,1+s_2),(1+s_1,1+s_2,1+s_3),(1+s_2,1+s_3,1+s_4)|>=0 for all real a,b,c.

In a G.P. if S_1 , S_2 & S_3 denote respectively the sums of the first n terms , first 2 n terms and first 3 n terms , then prove that, S_(1)^(2)+S_(2)^(2)=S_(1)(S_2+S_3) .

If S_(n)=1+1/2+1/3+…+1/n(ninN) , then prove that S_(1)+S_(2)+..+S_((n-1))=(nS((n))-n)or(nS((n-1))-n+1)

If S_1, S_2, S_3 are the sums to n, 2n, 3n terms respectively for a GP then prove that S_1,S_2-S_1,S_3-S_2 are in GP.

If the sum of n, 2n and infinite terms of G.P. are S_(1),S_(2) and S respectively, then prove that S_(1)(S_(1)-S)=S(S_(1)-S_(2)).

If S_(1),S_(2),S_(3) be respectively the sums of n,2n,3n terms of a G.P.,then prove that S_(1)^(2)+S_(2)^(2)=S_(1)(S_(2)+S_(3))

If alpha&beta be the real roots of ax^(2)+bx+c=0 and s_(n)=alpha^(n)+beta^(n) then prove that as_(n)+bs_(n-1)+cs_(n-2)=0 for all that n ge 2, n in N . Hence or otherwise prove that |{:(,3,1+s_(1),1+s_(2)),(,1+s_(1),1+s_(2),1+s_(3)),(,1+s_(2),1+s_(3),1+s_(4)):}|ge 0 for all real a,b,c

If S_(1),S_(2),S_(3) be respectively the sums of n,2n and 3n terms of a G.P.,prove that S_(1)(S_(3)-S_(2))=(S_(2)-S_(1))^(2)

If S_(1),S_(2) andS _(3) be respectively the sum of n,2n and 3n terms of a G.P.prove that S_(1)(S_(3)-S_(2))=(S_(2)-S_(1))^(2)