If for a G.P. S2=8 & S4=80, Find its first term and common ratio.

Question

Ans ...a=2, r=3....this is from G.P...

Answer 1

First term is a, S2=a+ar=8, S4=a+ar+ar^2+ar^3=80 where a is the first term and r the ratio of consecutive terms.

So a(1+r)=8 and a=8/(1+r); a(1+r+r^2+r^3)=80; 8(1+r+r^2+r^3)/(1+r)=80; 1+r+r^2+r^3=10+10r;

r^3+r^2-9r-9=0; r(r^2-9)+r^2-9=0; (r^2-9)(r+1)=0; (r-3)(r+3)(r+1)=0, so r=±3, -1. But a=8/(1+r) so r cannot be -1, otherwise a would be undefined. Therefore, r=-3, so a=-4; or r=3 and a=2.

CHECK: r=-3: -4+(-4)(-3)=8 and S4=8+(-4)(9)+(-4)(-27)=8-36+108=80. OK. And r=3: 2+6=8; 8+18+54=80. OK.

Answer 2

Sum to the nth term=a[1-rⁿ]/ (1-r)

Given . S2=8 & S4=80

S2 = a[1-r^2]/(1-r) = 8....(1)

S4 = a[1-r^4]/(1-r) = 80....(2)

By solving equation (1) and (2)

We have,
x= 2 an y =3

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