Question:
Let \( (a_n) \) be a sequence of positive real numbers such that
\[
a_1 = 1, \quad a_{n+1} = 2a_n a_{n+1} - a_n = 0 \text{ for all } n \geq 1.
\]
Then the sum of the series
\[
\sum_{n=1}^{\infty} a_n
\]
lies in the interval
Let \( (a_n) \) be a sequence of positive real numbers such that
\[
a_1 = 1, \quad a_{n+1} = 2a_n a_{n+1} - a_n = 0 \text{ for all } n \geq 1.
\]
Then the sum of the series
\[
\sum_{n=1}^{\infty} a_n
\]
lies in the interval
Show Hint
To solve recurrence relations, try to express the terms explicitly and look for patterns to find the sum of the series.
Updated On: Dec 11, 2025
- \( (1, 2) \)
- \( (2, 3) \)
- \( (3, 4) \)
- \( (4, 5) \)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the sequence.
The recurrence relation for \( a_n \) is \( a_{n+1} = 2a_n + a_{n+1} - a_n \). We solve for \( a_n \), and find the sum of the series.
Step 2: Conclusion.
Thus, the sum of the series lies between \( 1 \) and \( 2 \), so the correct answer is \( \boxed{(A)} \).
The recurrence relation for \( a_n \) is \( a_{n+1} = 2a_n + a_{n+1} - a_n \). We solve for \( a_n \), and find the sum of the series.
Step 2: Conclusion.
Thus, the sum of the series lies between \( 1 \) and \( 2 \), so the correct answer is \( \boxed{(A)} \).
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