Question

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

Answer 1

Step $\mathbf{1}$: Compute the value of $\mathit{n}$.

First term $a=6$

$T_n=216$ [ Last term ]

$$\begin{gathered} d=13-6 \\ =7 \end{gathered}$$ [ Common difference ]

Since,

$T_n=a+\left(n-1\right)d$

$$\begin{gathered} 216=6+(n-1)7 \\ 216-6=7n-7 \\ 210+7=7n \\ n=\frac{217}{7} \end{gathered}$$

$n=31$

Step $\mathbf{2}$: Compute the middle term.

Here the value of $n$ is odd. Hence,

Middle term $=\frac{n+1}{2}$th term

$$\begin{gathered} =\frac{31+1}{2} \\ =\frac{32}{2} \end{gathered}$$

$=16$ th term.

That is,

$T_{16}=6+(16-1)7$

$$\begin{gathered} =6+\left(15\times 7\right) \\ =111 \end{gathered}$$

Hence, the middle term of the AP is $\mathbf{111}$.