Question

Find the middle term of the AP 6, 13, 20, ..., 216. [CBSE 2015]

Answer 1

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,

$$\begin{gathered} a_n=216 \\ \Rightarrow 6+\left(n-1\right)\times 7=216\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow 7\left(n-1\right)=216-6=210 \\ \Rightarrow n-1=\frac{210}{7}=30 \end{gathered}$$
$\Rightarrow n=30+1=31$

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.