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**Directions for the below question:**

Let $a_1=p$ and $b_1 =q$ where $p$ and $q$ are positive quantities.

Define:

$a_n pb_{n-1} \: \: \: b_n=qb_{n-1}$ for even $n>1$

and $a_n pa_{n-1} \: \: \: b_n=qa_{n-1}$ for odd $n>1$

Which of the following best describes $a_n + b_n$ for even $n$?

- $q(pq)^{\frac{1}{2} n-1} (p+q)^{\frac{1}{2}n}$
- $q(pq)^{\frac{1}{2} n-1} (p+q)$
- $qp^{\frac{1}{2} n-1} (p+q)$
- $q^{\frac{1}{2} n} (p+q)$
- $q^{\frac{1}{2} n} (p+q)^{\frac{1}{2}n}$