Correct Answer - Option 2 : n/m
Concept:
If a1, a2, ..., an are in GP then common ratio is given by: r = ai + 1 / ai ∀ i =1, 2, …., n - 1.
The nth term in a GP is given by: an = arn - 1, where a is the first term and r is the common ratio.
Calculation:
Given: The sum of odd terms of the GP is m and sum of even terms of the GP is n.
\(\Rightarrow \frac{{Sum\;of\;even\;terms\;of\;GP}}{{Sum\;of\;odd\;terms\;of\;GP}} = \frac{{\left( {ar + a{r^3} + \ldots + a{r^{199}}} \right)}}{{\left( {a + a{r^2} + \ldots + a{r^{198}}} \right)}} = \frac{n}{m}\)
\(\Rightarrow \frac{{Sum\;of\;even\;terms\;of\;GP}}{{Sum\;of\;odd\;terms\;of\;GP}} = \frac{{ar \times \left( {1 + {r^2} + \ldots + {r^{198}}} \right)}}{{a \times \left( {1 + {r^2} + \ldots + {r^{198}}} \right)}} = \frac{n}{m}\)
⇒ r = n / m.