Question

If the sum of the first $2n$ terms of A.P $2,5,8,\dots$ is equal to the sum of the first $n$ terms of A.P $57,59,61,\dots .$, then $n$ is equal to:

• A. $10$
• B. $12$
• C. $11$
• D. $13$

Answer 1

The correct option is C

$11$

Explanation for the correct option.

Sum of $n$ terms of AP, ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{[}\mathbf{2}\mathbf{a}\mathbf{+}\left(n-1\right)\mathbf{d}\mathbf{]}$

Sum of $2n$ terms of AP $2,5,8,\dots$will be

$=\frac{2\mathrm{n}}{2}[2\left(2\right)+\left(2\mathrm{n}-1\right)3]$ [Here, $a=2,\mathrm{and}d=3$]

$=n\left(1+6n\right)$

Sum of $n$ terms of AP $57,59,61,\dots .$will be

$=\frac{\mathrm{n}}{2}[2\left(57\right)+\left(\mathrm{n}-1\right)2]$ [Here, $a=57,\mathrm{and}d=2$]

$$\begin{gathered} =\frac{n}{2}\left(112+2n\right) \\ =n\left(56+n\right) \end{gathered}$$

As per the given information,

$\begin{array}{rcl}n\left(1+6n\right)& =& n\left(56+n\right)\\ & \Rightarrow & 1+6n=56+n\\ \Rightarrow 5n& =& 55\\ \Rightarrow n& =& 11\end{array}$

Hence, option C is correct.