Question

The sum of first 20 terms of gp is 244 times the sum of first 10 terms. the common ratio is

Answer 1

Let the first term of the GP be $a$ and the common ration be $r$.

The sum of first $n$ terms of the GP is

$S_n=\frac{a(r^n-1)}{r-1}$.

It is given that

$S_{10}=\frac{a(r^{10}-1)}{r-1}\\ S_{20}=244S_{10}=\frac{a(r^{20}-1)}{r-1}\\ 244\frac{a(r^{10}-1)}{r-1}=\frac{a(r^{20}-1)}{r-1}\\ r^{20}-1=244(r^{10}-1)\\ r^{20}-244r^{10}+243=0\\ (r^{10}-1)(r^{10}-243)=0\\ r^{10}=1 \;or \; r^{10}=243$

Since $r\neq 1$,

$r^{10}=243\\ r^{10}=\sqrt{3} ^10\\ r=\sqrt{3}$

The common ratio is $\sqrt{3}$.