Question

Let $a_{n}$ be the n^th term of the G.P. of positive numbers. Let $\sum_{n = 1}^{100}{a_{2n} = \alpha}$ and $\sum_{n = 1}^{100}{a_{2n - 1} = \beta}$, such that α≠β, then the common ratio is

• A. $\frac{\alpha}{\beta}$
• B. $\frac{\beta}{\alpha}$
• C. $\sqrt{\frac{\alpha}{\beta}}$
• D. $\frac{\beta}{\alpha}$

Answer 1

Answer: (A)

$\frac{\alpha}{\beta}$

Answer 2

Let x be the first term and y, the common ratio of the G.P.

Then,$\alpha = \sum_{n = 1}^{100}{a_{2n} = a_{2} + a_{4} + a_{6} + .... + a_{200}}$and $\beta = \sum_{n = 1}^{100}{a_{2n - 1} = a_{1} + a_{3} + a_{5} + ...... + a_{199}}$

⇒$\alpha = xy + xy^{3} + xy^{5} + ..... + xy^{199} = xy\frac{1 - (y^{2})^{100}}{1 - y^{2}} = xy\left( \frac{1 - y^{200}}{1 - y^{2}} \right)\beta = x + xy^{2} + xy^{4} + ..... + xy^{198} = x \cdot \frac{1 - (y^{2})^{100}}{1 - y^{2}} = x \cdot \left( \frac{1 - y^{200}}{1 - y^{2}} \right)$∴$\frac{\alpha}{\beta} = y$. Thus, common ratio $= \frac{\alpha}{\beta}$