Question:
The sum of $n$ terms of an $A P$ is $\frac{1}{2}$ an $^{2}+b n$. Find the common difference.
Solution:
Given: the sum of $n$ terms of an AP is $\frac{1}{2} a n^{2}+b n$
To find: common difference.
Put n = 1 we get
First term $=\frac{1}{2}+b$
Put n = 2 we get
First term + second term = 2 × a + 2 × b
Second term $=\frac{3}{2} a+b$
Therefore common difference will be
Second term - first term
Common difference $=2 \mathrm{a}$