Answer to Question #292235 in Algebra for Spitchomz

Question

The first term of a geometric series is 9 and the ratio of the sum of the first eight terms to the sum of the first four terms is 97:81.

Calculate the first 3 terms of the sequence it is given that all the terms are positive.

Answer 1

The sum of first n terms of a geometric progression with first term $a$ is,$~S_{n} = \dfrac{a(r^{n}-1)}{r - 1}, r\ne1.$

Here given

$a = 9, S_{8}:S_{4}=97:81$.

Therefore,

$\dfrac{S_{8}}{S_4} = \dfrac{\frac{9(r^{8}-1)}{r - 1}}{\frac{9(r^{4}-1)}{r - 1}}= \dfrac{9(r^{8}-1)}{r - 1}\cdot \dfrac{r - 1}{9(r^{4}-1)}\\ \dfrac{97}{81}=\dfrac{r^{8}-1}{r^{4}-1}\\ \dfrac{97}{81}=\dfrac{(r^{4})^2-1}{r^{4}-1}\\ \dfrac{97}{81}=\dfrac{(r^{4}-1)(r^{4}+1)}{r^{4}-1}=r^4 +1\\ \dfrac{97}{81}-1=r^4\\ \dfrac{16}{81}=r^4\\ (\frac{2}{3})^4=r^4\\ \therefore r=\frac{2}{3}$

Hence the first three terms are $a, ar, ar^2$ , that is 9,6,4.