Question

Find the sum of $p$ terms of the series whose $n^{th}$ term is $\frac{n}{a}+b$.

Answer 1

Let the first term of a series be $x$ and common difference be $d$

The $n$th term of an Arithmetic progression series is $x+(n-1)d=x-d+nd$

Given that the $n$th term in the series is $\frac{n}{a}+b$

By comparing , we get $d=\frac{1}{a}$ and $x=b+\frac{1}{a}$

Now the sum of $p$ terms of the series is $\frac{p}{2}(2x+(p-1\left)d\right)=\frac{p}{2}(2b+\frac{2}{a}+\frac{p-1}{a})=\frac{p}{2}(2b+\frac{p+1}{a})$