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For any increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, ......$ has the property that for every positive integer k, the subsequence $a_{2 k - 1}, a_{2 k}, a_{2 k + 1}$ is in geometry sequence and the subsequence $a_{2 k}, a_{2 k + 1}, a_{2 k + 2}$ is in Arithmetic sequence, if $a_{13} = 2016$ then $a_{1}$ is ________

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answer is 504.

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Detailed Solution

Let $a_{1} = a$ and $a_{2} = k a$
Sequence becomes $a, k a, k^{2} a, k^{2} a, k (2 k - 1) a, {(2 k - 1)}^{2} a, (2 k - 1) (3 k - 2) a, {(3 k - 2)}^{2} a$
$\Rightarrow a = \frac{2016}{{(6 k - 5)}^{2}} = \frac{2016}{2^{2}} = 504$

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