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?

• A. \( X_n \) is positive if \( n \) is even
• B. \( X_n \) is positive if \( n \) is odd
• C. \( X_n \) is negative if \( n \) is even
• D. None of these

Hint:

When given recurrence relations, look for patterns in the sequence to simplify solving.

Answer 1

The Correct Option is A

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.