Question

THE SUM OF FIRST 20 TERMS OF A G.P. IS 244 TIMES THE SUM OF ITS FIRST 10 TERMS. the comman ratio is?

Answer 1

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 $r^{10}=t$

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

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

[tex] Common\ ratio\ is\ 1\ and\ \sqrt{3} .[/tex]