The sum of first q terms …

The question is wrong. It  should be 9 in place of q

Let first term of AP is a And Common difference is d.

Given:

{tex}a_6 = a+5d\\a_{13}=a+12d{/tex}

{tex}{a_6\over a_{13}}={1\over 2}{/tex}

{tex} =>{ a+5d\over a+12d}={1\over 2}{/tex}

{tex}=> 2a+ 10 d = a+12d {/tex}

{tex}=> a = 2d{/tex}

Also, 

{tex}S_9= {9\over 2}[2a+(9-1)d]{/tex}

{tex}=> 162 = {9\over 2}(2a+8d){/tex}

{tex}=> 36=[4d+8d]{/tex}

{tex}=> d = 3 {/tex}

Then a = 6

{tex}a_{15}= 6+14\times 3 = 48{/tex}

{tex}{/tex}

The data is insufficient. The q should be replaced with 9.

Let first term of AP = a

Common difference = d

{tex}a_6 = a+5d\\a_{13}=a+12d{/tex}

According to question,

{tex}{a_6\over a_{13}}={1\over 2}{/tex}

{tex} =>{ a+5d\over a+12d}={1\over 2}{/tex}

=> 2a+ 10 d = a+12d

=> a = 2d ... (1)

Also, 

{tex}S_9=> 162= {9\over 2}[2a+8d]{/tex}

{tex}=> 36=[4d+8d]{/tex}

=> d = 3

So, a = 6

{tex}a_{15}= 6+14\times 3 = 48{/tex}

{tex}{/tex}