Question:
The nth element of a series is represented as
\[
X_n = (-1)^{n} X_{n-1}.
\]
If \( X_0 = x \) and \( x>0 \), then which of the following is always true?
The nth element of a series is represented as
\[
X_n = (-1)^{n} X_{n-1}.
\]
If \( X_0 = x \) and \( x>0 \), then which of the following is always true?
Show Hint
When given recurrence relations, look for patterns in the sequence to simplify solving.
Updated On: Mar 20, 2026
- \( X_n \) is positive if \( n \) is even
- \( X_n \) is positive if \( n \) is odd
- \( X_n \) is negative if \( n \) is even
- None of these
Show Solution
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The Correct Option is A
Solution and Explanation
The recurrence relation \( X_n = (-1)^n X_{n-1} \) implies that the value of \( X_n \) alternates in sign with each successive term:
- For even \( n \), \( X_n = x \).
- For odd \( n \), \( X_n = -x \).
Thus, \( X_n \) is positive when \( n \) is even.
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