Concept 5.1

What Makes a List an AP

Source: NCERT Mathematics, Class 108 minute lesson

Formal statement

An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first term. This fixed number is called the common difference, d. It can be positive, negative or zero.

The intuition

Don't look at the numbers. Look at the spaces between them. 3, 7, 11, 15 is an AP not because it grows, but because it grows by the same amount every time.

That one fixed step is the whole personality of an AP. Know the first term and the step, and you know every term in the list, however far it goes.

The attention

Look at the gaps between the terms, not at the terms.

3+47+411+415+419

Look hereNot here

Look here: The gaps between terms. An AP is defined by how it changes. If every gap is the same, it is an AP, and that gap is d.
Not here: The terms themselves. Growing numbers feel like a pattern. But 1, 2, 4, 8 grows too, and its gaps are 1, 2, 4. Not an AP.
What it teaches: To recognise a structure, look at what stays the same, not at what changes.