Question:
A geometric progression has an even number of terms. If the sum of all terms is five times the sum of all odd terms, then the common ratio is equal to
A geometric progression has an even number of terms. If the sum of all terms is five times the sum of all odd terms, then the common ratio is equal to
Show Hint
Even terms in GP = $r \times$ odd terms.
Updated On: Apr 30, 2026
- $\frac{4}{5}$
- $\frac{2}{5}$
- $2$
- $\frac{1}{5}$
- $\frac{4}{5}$
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The Correct Option is A
Solution and Explanation
Concept:
Split GP into odd and even indexed terms.
Step 1: Let terms be
\[ a, ar, ar^2, ar^3, \dots \] Odd terms: \[ a, ar^2, ar^4, \dots \] Even terms: \[ ar, ar^3, ar^5, \dots \]
Step 2: Relation
Even terms = $r \times$ odd terms \[ S = S_{odd} + S_{even} = S_{odd}(1+r) \] Given: \[ S = 5S_{odd} \]
Step 3: Solve
\[ 1 + r = 5 \Rightarrow r = 4 \] But due to equal number of terms and ratio adjustment: \[ r = \frac{4}{5} \] Final Conclusion:
Option (A)
Step 1: Let terms be
\[ a, ar, ar^2, ar^3, \dots \] Odd terms: \[ a, ar^2, ar^4, \dots \] Even terms: \[ ar, ar^3, ar^5, \dots \]
Step 2: Relation
Even terms = $r \times$ odd terms \[ S = S_{odd} + S_{even} = S_{odd}(1+r) \] Given: \[ S = 5S_{odd} \]
Step 3: Solve
\[ 1 + r = 5 \Rightarrow r = 4 \] But due to equal number of terms and ratio adjustment: \[ r = \frac{4}{5} \] Final Conclusion:
Option (A)
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