Question

Which of the following best describes \( a_n + b_n \) for even \( n \)?

• A. \( q (pq)^{\frac{n}{2} - 1} (p + q) \)
• B. \( qp^{\frac{n}{2} - 1}(p + q) \)
• C. \( q^{\frac{n}{2}}(p + q) \)
• D. \( q^{\frac{n}{2}}(p + q)^{\frac{n}{2}} \)
• E. \( q (pq)^{\frac{n}{2} - 1}(p + q)^{\frac{n}{2}} \)

Hint:

Calculate first few terms to identify exponent patterns, then generalize.

Answer 1

The Correct Option is A

We trace the sequence: For \( n = 2 \) (even): \( a_2 = p b_1 = pq \), \( b_2 = q b_1 = q^2 \), sum = \( pq + q^2 = q(p + q) \).
For \( n = 4 \):
From \( n=3 \) (odd): \( a_3 = p a_2 = p^2 q \), \( b_3 = q a_2 = pq^2 \).
Then \( a_4 = p b_3 = p^2 q^2 \), \( b_4 = q b_3 = p q^3 \), sum = \( p^2 q^2 + p q^3 = p q^2 (p+q) \).
Pattern:
For even \( n \), \( a_n + b_n = q(pq)^{\frac{n}{2} - 1}(p+q) \).