[Math] Find the common ratio, if the sum of first 8 terms in GP(geometric progression) is 5 times the sum of first 4 terms also in GP.

Question

I tried it by supposing, (the supposition where “if the number of terms is odd, we take the middle term as ‘a’ and the common ratio as ‘r’. If the number of terms is even, we take ‘a\r’ and ‘ar’ as the middle terms and ‘r2 ‘as the common ratio.”
Sum of first 8 terms in GP,
S8 = a/r7 + a/r5 + a/r3 + a/r + ar + ar3 + ar5+ ar7
S4 = a/r3 + a/r + ar + ar3
by question,
S8 = 5S4 …………………………equ(1)
I tried solving equ(1) for r2
But still couldn’t find out the answer.
Please give it a try!!!!!!

Answer 1

If the first term & the common ratio be $a,r\ne1$ respectively,

the sum of $n(\ge1)$ terms $$S_n=\frac{a(r^n-1)}{r-1}$$

So, $$5=\frac{S_8}{S_4}=\frac{r^8-1}{r^4-1}=r^4+1$$