# Write the common difference of an A.P. the sum of whose first

Question:

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

Solution:

Sum of the first $n$ terms of an A.P. $=\frac{p}{2} n^{2}+Q n$

Sum of one term of an A.P. $=S_{1}$

$\Rightarrow \frac{p}{2}(1)^{2}+Q(1)$

$\Rightarrow \frac{p}{2}+Q$

Sum of two terms of an A.P. $=S_{2}$

$\Rightarrow \frac{p}{2}(2)^{2}+Q(2)$

$\Rightarrow 2 p+2 Q$

Now, we have:

$a_{1}+a_{2}=S_{2}$

$\Rightarrow \frac{p}{2}+Q+a_{2}=2 p+2 Q$

$\Rightarrow a_{2}=Q+\frac{3}{2} p$

Common difference:

$d=a_{2}-a_{1}$

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

$=p$