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

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

### Question

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

### Options

A) $$\sqrt{2}$$

B) $$\sqrt{5}$$

C) 3

D) 5

### Explanation:

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

## Dicussion (1)

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