Question

Find the middle term of the A.P. $6,13,20,....,216$.

Answer 1

Clearly, $6,13,20,.....,216$ is an A.P. with first term $a=6$ and common difference $d=7$.

Let there be $n$ terms in the given A.P.

Then, $a_n=216$

$⟹a+(n+1)d=216$

$⟹6+7(n-1)=216$ $[\because a=6$ and $d=7]$

$⟹7n=217⟹n=31$
Here, $n$ is odd so ${\left(\frac{n+1}{2}\right)}^{th}$ i.e. ${\left(\frac{31+1}{2}\right)}^{th}={16}^{th}$ term is the middle term and is given by

$a_{16}=a+(16-1)d=1+15d=6+15\times 7=111$