Question

An AP consists of $23$terms. If the sum of the three terms in the middle is $141.$ and the sum of the last three terms is$261$, then the first term is

• A. $6$
• B. $5$
• C. $4$
• D. $3$

Answer 1

The correct option is D

$3$

The explanation for the correct option:

Step 1. Assume the variables :

Let $a_1,a_2.........,a_{23}$be the terms in AP.

Let $d$be the common difference.

Step 2. Apply the given condition :

the sum of the three terms in the middle is $141.$

$$\begin{gathered} \Rightarrow a_{11}+a_{12}+a_{13}=141 \\ \Rightarrow a+10d+a+11d+a+12d=141\mathbf{\left[}\mathbf{\because }\mathbf{}{\mathbf{a}}_{\mathbf{n}}\mathbf{=}\mathbf{}\mathbf{a}\mathbf{}\mathbf{+}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{d}\mathbf{\right]} \\ \Rightarrow 3a+33d=141 \\ \Rightarrow a+11d=47..............\left(1\right) \end{gathered}$$

the sum of the last three terms is $261.$

$$\begin{gathered} \Rightarrow a_{21}+a_{22}+a_{23}=261 \\ \Rightarrow a+20d+a+21d+a+22d=261 \\ \Rightarrow 3a+63d=261 \\ \Rightarrow a+21d=87.............\left(2\right) \end{gathered}$$

Step3. Solving the equation :

Solving $\left(1\right)$ and $\left(2\right)$, we get $d=4$

Substitute $d$in $\left(1\right)$, we get

$$\begin{gathered} \Rightarrow a+44=47 \\ \Rightarrow a=3 \end{gathered}$$

Thus, the first term is $3$.

Hence, the correct option is D.