Question

If the $\mathrm{nth}$ term of an AP is $(2\mathrm{n}+1)$, then find the sum of the first $\mathrm{n}$ terms of the AP.

Answer 1

Step 1: Find the first term and common difference of AP

Given: $nth$ term of AP is given as $(2\mathrm{n}+1)$.

$$\begin{gathered} {\mathrm{a}}_1=2\times 1+1=3 \\ {\mathrm{a}}_2=2\times 2+1=5 \\ {\mathrm{a}}_3=2\times 3+1=7 \end{gathered}$$

$3,5,7,.....$ are the terms of AP

Hence $\mathrm{a}=3\mathrm{and}\mathrm{d}=2$

Step 2: Find the sum of AP

$$\begin{gathered} {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2\mathrm{a}+\left(\mathrm{n}-1\right)\mathrm{d}] \\ {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[2\times 3+(\mathrm{n}-1)2\right] \\ {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[6+2\mathrm{n}-2\right] \\ {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[2\mathrm{n}+4\right] \end{gathered}$$

Hence the sum of $\mathrm{n}$ terms of AP is ${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2\mathrm{n}+4]$.