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Sum of first n terms of a GP is given by
S_n= \frac{a(1-r^n)}{1-r}

 \frac{S_{20}}{S_{10}} =244\\ \\ \Rightarrow \frac{a(1-r^{20})}{1-r}\ \div\ \frac{a(1-r^{10})}{1-r} =244\\ \\ \Rightarrow \frac{a(1-r^{20})}{1-r}\ \times\ \frac{1-r}{a(1-r^{10})} =244\\ \\ \Rightarrow  \frac{1-r^{20}}{1-r^{10}} =244\\ \\ \Rightarrow 1-r^{20}=244 \times (1-r^{10})\\ \\ \Rightarrow 1-r^{20}=244-244r^{10}\\ \\ \Rightarrow r^{20}-244r^{10}=1-244=-243\\ \\ \Rightarrow r^{20}-244r^{10}+243=0

Take r^{10}=t

t^2-244t+243=0\\ \\ \Rightarrow t^2-243t-t+243=0\\ \\ \Rightarrow t(t-243)-1(t-243)=0\\ \\ \Rightarrow (t-1)(t-243)=0\\ \\ \Rightarrow t=1\ or\ 243

Thus\ r^{10}=1\\ or\ r=\boxed1\\ \\r^{10}=243=3^5\\or\ r=3^{ \frac{5}{10} }\\or\ r = \boxed{ \sqrt{3} }

Common\ ratio\ is\ 1\ and\  \sqrt{3} .

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