What is the common ratio of the G.P. (√10+√5)+(√10+2√5)+... ( 10 + 5 ) + ( 10 + 2 5 ) + . . . ?

What is the common ratio of the G.P. $(\sqrt{10}+\sqrt{5})+(\sqrt{10}+2\sqrt{5})+...$?

Answer Details

Common ratio r of the G.P is
$r=\frac{T_n+1}{T_n}=\frac{T_2}{T_1}$
$r=\frac{\sqrt{10}+2\sqrt{5}}{\sqrt{10}+\sqrt{5}}$
$r=\frac{\sqrt{10}+2\sqrt{5}}{\sqrt{10}+\sqrt{5}}\times \frac{\sqrt{10}-\sqrt{5}}{\sqrt{10}-\sqrt{5}}$
$=\frac{\left(\sqrt{10}\right)\left(\sqrt{10}\right)+\left(\sqrt{10}\right)(-\sqrt{5})+\left(2\sqrt{5}\right)\left(\sqrt{10}\right)+\left(2\sqrt{5}\right)(-\sqrt{5})}{(\sqrt{10}{)}^2-(\sqrt{5}{)}^2}$
$\frac{10-\sqrt{50}+2\sqrt{50}-10}{10-5}$
$\frac{\sqrt{50}}{5}$
$\frac{\sqrt{25\times 2}}{5}$
$\frac{5\sqrt{2}}{5}$
$\sqrt{2}$