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) 5−4
(B) 51
(C) 4
(D) none the these
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Step-by-Step Solution
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,ar2,…,ar2n−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: S2n=r−1a(r2n−1).
Step 3: Formulate the sum of odd terms
The odd terms are a,ar2,ar4,…,ar2n−2. This is also a G.P. with first term a, common ratio r2, and n terms. The sum of these odd terms is Sodd=r2−1a((r2)n−1)=r2−1a(r2n−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 S2n and Sodd into this equation.
Step 5: Solve for the common ratio
We can cancel out the common term a(r2n−1) from both sides, assuming a=0 and r2n−1=0. Then, we use the identity r2−1=(r−1)(r+1) to simplify the equation. This allows us to solve for r.