Sum of AP

1. Introduction: Adding a Long List Without Adding It

You already know what an AP is. An AP — Arithmetic Progression — is a list of numbers where each term jumps by the same fixed amount. That fixed jump is the common difference, written $d$. The first term is written $a$.

So $3, 7, 11, 15, \ldots$ is an AP with first term $a = 3$ and common difference $d = 4$.

Now here is the new question. Suppose someone asks you to add up the first 100 terms of an AP. Not find the 100th term — add all 100 of them. Are you going to write them all out and add one by one? That would take an hour, and you would slip somewhere in the middle.

This lesson gives you a way to add a long AP in three or four lines, no matter how many terms there are. The total of the terms of an AP is called its sum, written $S_n$ — the sum of the first $n$ terms. The small $n$ just tells you how many terms you are adding.

Stop scrolling. Try it in your head before reading on. The AP $2, 4, 6, 8, 10$ — what is the total if you just add them? Hold that number; we will come back to it.

By the end, you will be handed any AP and a number of terms, and you will produce the total calmly, using one formula you trust.

You can now say what this lesson is about: finding the total of many terms of an AP quickly, instead of adding them one at a time.