Question

The $n^{th}$ term of the series $2+4+7+....$ will be

• A. $\frac{n^2+n+1}{2}$
• B. $n^2+n+2$
• C. $\frac{n^2+n+2}{2}$
• D. $\frac{n^2+2n+2}{2}$

Answer 1

The correct option is C

$\frac{n^2+n+2}{2}$

Find the given term of the arithmetic progression

Given series is $2+4+7+....$

$$\begin{gathered} a_1=2 \\ {a}_2=4=a_1+2=2+2 \\ {a}_3=7=a_2+3=2+2+3 \\ {a}_4=11=a_3+4=2+2+3+4 \end{gathered}$$

So, adding the last terms,

$$\begin{gathered} a_n=2+2+3+4+..........+n \\ =1+\left(1+2+3+4+..........+n\right) \\ =1+\frac{n(n+1)}{2}\left[\because \mathrm{Sum}\mathrm{of}\mathrm{first}\mathrm{n}\mathrm{natural}\mathrm{number}\mathrm{is}\frac{n(n+1)}{2}\right] \\ =\frac{n^2+n+2}{2} \end{gathered}$$

Hence, the correct option is $\left(\mathrm{C}\right)$.