Find the middle term of the AP 6, 13, 20, ..., 216.

The given AP is 6, 13, 20, ..., 216. 

First term, a = 6

Common difference, d = 13 − 6 = 7

Suppose there are n terms in the given AP. Then,

$a_{n}=216$

$\Rightarrow 6+(n-1) \times 7=216 \quad\left[a_{n}=a+(n-1) d\right]$

$\Rightarrow 7(n-1)=216-6=210$

$\Rightarrow n-1=\frac{210}{7}=30$

$\Rightarrow n=30+1=31$

Thus, the given AP contains 31 terms.

∴ Middle term of the given AP

$=\left(\frac{31+1}{2}\right)$ th term

= 16th term

= 6 + (16 − 1) × 7

= 6 + 105

= 111
 
Hence, the middle term of the given AP is 111.