If the sum of first five terms of a G.P. is 31 and the sum of its first ten terms is 1023, then the G.P. is

$S_n=a\cdot \frac{r^n-1}{r-1}$

$S_5=a\cdot \frac{r^5-1}{r-1}=31$

$S_{10}=a\cdot \frac{r^{10}-1}{r-1}=1023$

$\frac{S_{10}}{S_5}=\frac{r^{10}-1}{r^5-1}=\frac{1023}{31}$

$=\frac{(r^5+1)(r^5-1)}{r^5-1}=33$

$\Rightarrow r^5+1=33$

$\Rightarrow r^5=33-1=32$

$\Rightarrow r^5=2^5$

$\Rightarrow r=2$

$\Rightarrow S_5=31=a\cdot \frac{r^5-1}{r-1}\Rightarrow a\cdot \frac{32-1}{2-1}\Rightarrow a\cdot 31\Rightarrow 31\Rightarrow a=1$

$\therefore$ G.P. is $\{1,2,4,8,\cdots \}$