Common ratio r of the G.P is

\(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\)

\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\)

\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \)

\( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\)

\(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\)

\(\frac{\sqrt{50}}{5}\)

\(\frac{\sqrt{25 \times 2}}{5}\)

\(\frac{5\sqrt{2}}{5}\)

\(\sqrt{2}\)