Nth Term of AP

1. Introduction: Reaching a Far-Off Term Without Counting

You already know what an Arithmetic Progression is. It is a list of numbers where you add the same fixed number each time to get the next one. That fixed number is the common difference.

Here is a small one: $3, 7, 11, 15, \ldots$ Each time you add 4.

Now suppose someone asks: "What is the 50th number in this list?"

You could write them all out — $3, 7, 11, 15, 19, \ldots$ — adding 4 again and again until you reach the 50th. But that is 49 additions. One slip anywhere and the whole answer is wrong.

There has to be a faster way. A way to jump straight to the 50th term without walking through the other 49.

Stop scrolling. Try it in your head before reading on: in $3, 7, 11, 15, \ldots$, how many times do you add 4 to get from the 1st term to the 4th term?

Three times. Not four. To reach the 4th term you start at the 1st and add 4 three times: $3 \to 7 \to 11 \to 15$. Hold on to that idea — the jump count is one less than the term number. That single fact is the whole engine of this lesson.

By the end of this lesson you can find any term of any AP — the 50th, the 100th, the 200th — with one short calculation instead of a long list.