In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is (A) $\frac{-4}{5}$ (B) $\frac{1}{5}$ (C) 4 (D) none the these

Step 1: Define the Geometric Progression

Let the geometric progression (G.P.) have $2n$ terms, where $a$ is the first term and $r$ is the common ratio. The terms of the G.P. can be written as $a, ar, ar^2, \dots, ar^{2n-1}$.

Step 2: Formulate the sum of all terms

The sum of all $2n$ terms of the G.P. is given by the standard formula for the sum of a geometric series: $S_{2n} = \frac{a(r^{2n}-1)}{r-1}$.

Step 3: Formulate the sum of odd terms

The odd terms are $a, ar^2, ar^4, \dots, ar^{2n-2}$. This is also a G.P. with first term $a$, common ratio $r^2$, and $n$ terms. The sum of these odd terms is $S_{\text{odd}} = \frac{a((r^2)^n - 1)}{r^2-1} = \frac{a(r^{2n}-1)}{r^2-1}$.

Step 4: Set up the given condition

According to the problem statement, the sum of all terms is 5 times the sum of the odd terms. We substitute the expressions for $S_{2n}$ and $S_{\text{odd}}$ into this equation.

Step 5: Solve for the common ratio

We can cancel out the common term $a(r^{2n}-1)$ from both sides, assuming $a \neq 0$ and $r^{2n}-1 \neq 0$. Then, we use the identity $r^2-1 = (r-1)(r+1)$ to simplify the equation. This allows us to solve for $r$.

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