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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 $n$?

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