Question

Write the common difference of an A.P. the sum of whose first n terms is $\frac{p}{2}{n}^2+Qn$.

Answer 1

Sum of the first n terms of an A.P. = $\frac{p}{2}{n}^2+Qn$
Sum of one term of an A.P. = $S_1$
$$\begin{gathered} \Rightarrow \frac{p}{2}{\left(1\right)}^2+Q\left(1\right) \\ \Rightarrow \frac{p}{2}+Q \end{gathered}$$

Sum of two terms of an A.P. = $S_2$
$$\begin{gathered} \Rightarrow \frac{p}{2}{\left(2\right)}^2+Q\left(2\right) \\ \Rightarrow 2p+2Q \end{gathered}$$
Now, we have:
$$\begin{gathered} a_1+a_2=S_2 \\ \Rightarrow \frac{p}{2}+Q+a_2=2p+2Q \\ \Rightarrow a_2=Q+\frac{3}{2}p \end{gathered}$$

Common difference:
$$\begin{gathered} d=a_2-a_1 \\ =\left(Q+\frac{3}{2}p\right)-\left(Q+\frac{p}{2}\right) \\ =p \end{gathered}$$