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

1. Analyze the Question

We are given the first two terms of a Geometric Progression (G.P.):

• First term ($a$): $\sqrt{10} + \sqrt{5}$
• Second term ($ar$): $\sqrt{10} + 2\sqrt{5}$

We need to find the common ratio ($r$).

2. Explain the Concept

In a Geometric Progression, the common ratio ($r$) is found by dividing any term by the term immediately preceding it:

3. Show the Solution

Given: $a = \sqrt{10} + \sqrt{5}$ $ar = \sqrt{10} + 2\sqrt{5}$

Substitute these into the formula:

To simplify this, we can factor out $\sqrt{5}$ from both the numerator and the denominator, since $\sqrt{10} = \sqrt{5} \times \sqrt{2}$:

Numerator: $\sqrt{10} + 2\sqrt{5} = \sqrt{5}(\sqrt{2} + 2)$ Denominator: $\sqrt{10} + \sqrt{5} = \sqrt{5}(\sqrt{2} + 1)$

Now, divide them:

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator $(\sqrt{2} - 1)$:

Expand the numerator: $(\sqrt{2} \times \sqrt{2}) - (\sqrt{2} \times 1) + (2 \times \sqrt{2}) - (2 \times 1)$ $= 2 - \sqrt{2} + 2\sqrt{2} - 2$ $= \sqrt{2}$

Expand the denominator: $(\sqrt{2})^2 - (1)^2$ $= 2 - 1 = 1$

So, $r = \frac{\sqrt{2}}{1} = \sqrt{2}$

4. Identify the Answer

The calculated common ratio is $\sqrt{2}$, which matches option A.

5. Exam Tip

When you see square roots in a G.P. question, look for common factors immediately. Factoring out the smallest root (in this case, $\sqrt{5}$) often simplifies the expression significantly before you even need to perform complex rationalization.