A geometric progression (GP) consists of 200 terms. If the sum of odd terms of the GP is m, and the sum of even terms of the GP is n, then what is its common ratio?

Concept:

If a₁, a₂, ….., a_n are in GP then common ratio is given by: r = a_{i + 1} / a_i ∀ i =1, 2, …., n - 1.

The n^th term in a GP is given by: a_n = ar^{n - 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.

\(\Rightarrow \frac{{Sum\;of\;even\;terms\;of\;GP}}{{Sum\;of\;odd\;terms\;of\;GP}} = \frac{{\left( {ar + a{r^3} + \ldots + a{r^{199}}} \right)}}{{\left( {a + a{r^2} + \ldots + a{r^{198}}} \right)}} = \frac{n}{m}\)

\(\Rightarrow \frac{{Sum\;of\;even\;terms\;of\;GP}}{{Sum\;of\;odd\;terms\;of\;GP}} = \frac{{ar \times \left( {1 + {r^2} + \ldots + {r^{198}}} \right)}}{{a \times \left( {1 + {r^2} + \ldots + {r^{198}}} \right)}} = r = \frac{n}{m}\)