An arithmetic series is what we get when we add together the terms in an arithmetic sequence.

Suppose we have the arithmetic sequence

Then we can check that, for example,

However, working out every term in the sequence and adding them together is boring. In fact, there is a formula

You need to remember the proof for this formula. We'll discuss the proof in the example below, and you'll be asked to do it for yourself as one of the problems.

1. The arithmetic sequence $u_n$ has first term $100$ and common difference $-3.$

   Find the sum of the first $20$ terms of this sequence.

2. Find the value of the arithmetic series

   $$3+\frac{1}{2}-2-\dots -52$$

3. Find the value of

   $$\sum _{k=4}^{30}60-5k$$