Predicting What Comes Next: Exploring Sequences and Progressions

A sequence is an ordered list of numbers that follow a particular rule or pattern. Each number is called a term, and its position matters. For example, 2, 4, 6, 8, ... is a sequence formed by adding 2 each time.

An arithmetic progression (AP) is a sequence in which the difference between any two consecutive terms is the same. This fixed difference is called the common difference. For example, 5, 8, 11, 14, ... is an AP with a common difference of 3.

Subtract any term from the term that comes right after it. If this difference is the same throughout the sequence, it is the common difference of an arithmetic progression. For 7, 10, 13, ..., the common difference is 10 − 7 = 3.

First identify the rule that links the terms — for example, adding a fixed number or multiplying by a factor. Then apply that rule to the last known term. In an arithmetic progression you simply add the common difference to the previous term.

The general term, often written as the nth term, is a formula that gives any term directly from its position number n. For an arithmetic progression it is a + (n − 1)d, where a is the first term and d is the common difference, so you can find a far-off term without listing them all.

A finite sequence has a definite number of terms and comes to an end, such as 1, 2, 3, 4, 5. An infinite sequence continues forever without a last term, shown with dots like 1, 2, 3, 4, ... .