#### Question

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

#### Solution

Let the G.P. be T_{1}, T_{2}, T_{3}, T_{4}, … T_{2}_{n}.

Number of terms = 2*n*

According to the given condition,

T_{1} + T_{2} + T_{3} + …+ T_{2}_{n} = 5 [T_{1} + T_{3} + … +T_{2}_{n}_{–1}]

⇒ T_{1} + T_{2} + T_{3} + … + T_{2}_{n} – 5 [T_{1} + T_{3} + … + T_{2}_{n}_{–1}] = 0

⇒ T_{2} + T_{4} + … + T_{2}_{n} = 4 [T_{1} + T_{3} + … + T_{2}_{n}_{–1}]

Let the G.P. be *a*, *ar*, *ar*^{2}, *ar*^{3}, …

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Solution A G.P. Consists of an Even Number of Terms. If the Sum of All the Terms is 5 Times the Sum of Terms Occupying Odd Places, Then Find Its Common Ratio. Concept: Geometric Progression (G. P.).