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

If S=1+a+a^2+a^3+a^4+……….to oo then a=

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

If the sum of first n,2n,3n terms of an A.P. be S_1 , S_2 and S_3 respectively then prove that S_3=3(S_2-S_1)

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 S_1 , S_2 and S_3 are respectively the sum of n, 2n and 3n terms of a G.P., then prove that S_1(S_3 - S_2) = (S_2 - S_1)^2 .

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.

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 the sum of first n,2n,3n terms of an A.P. be S_1 , S_2 and S_3 respectively then prove that S_1+S_3=2S_2

If 2s = a + b + c , Then prove that (s-a)^3+(s-b)^3+(s-c)^3-3(s-a)(s-b)(s-c) = 1/2(a^3+b^3+c^3-3abc)