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. = \[S_1\]

\[\Rightarrow \frac{p}{2} \left( 1 \right)^2 + Q\left( 1 \right)\]

\[ \Rightarrow \frac{p}{2} + Q\]

Sum of two terms of an A.P. =

\[S_2\]

\[\Rightarrow \frac{p}{2} \left( 2 \right)^2 + Q\left( 2 \right)\]

\[ \Rightarrow 2p + 2Q\]

Now, we have:

\[a_1 + a_2 = S_2 \]

\[ \Rightarrow \frac{p}{2} + Q + a_2 = 2p + 2Q\]

\[ \Rightarrow a_2 = Q + \frac{3}{2}p\]

Common difference:

\[d = a_2 - a_1 \]

\[ = \left( Q + \frac{3}{2}p \right) - \left( Q + \frac{p}{2} \right)\]

\[ = p\]