Question

There are $15$ terms in an arithmetic progression. Its first term is $5$ and their sum is $390$. The middle term is

• A. $23$
• B. $26$
• C. $29$
• D. $32$

Answer 1

The correct option is B

$26$

Find the middle term of the given arithmetic progression

Given: The first term$=5$, the number of terms$=15$, and the sum of terms $=390$

Sum of term is given by ${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[2\mathrm{a}+\left(\mathrm{n}-1\right)\mathrm{d}\right]$

$$\begin{gathered} \Rightarrow 390=\frac{15}{2}\left[2\times 5+\left(15-1\right)\mathrm{d}\right] \\ \Rightarrow 390=\frac{15}{2}\left[10+14\mathrm{d}\right] \\ \Rightarrow 390=15\left[5+7\mathrm{d}\right] \\ \Rightarrow 390=75+105\mathrm{d} \\ \Rightarrow 390-75=105\mathrm{d} \\ \Rightarrow 315=105\mathrm{d} \\ \Rightarrow \frac{315}{105}=\mathrm{d} \\ \Rightarrow 3=\mathrm{d} \end{gathered}$$

The mid-term of the AP is the $8\mathrm{th}$ term $=\mathrm{a}+7\mathrm{d}$

$$\begin{gathered} =5+7\times 3 \\ =5+21 \\ =26 \end{gathered}$$

Hence, option (B) is correct