Sequences and their terms

A sequence is an ordered list of numbers; each number is a term. In the square numbers 1, 4, 9, 16, 25, …, 1 is the first term and 25 the fifth.

• the dots '…' mean it goes on for ever: an infinite sequence;
• 6, 12, 24, 48, 96 stops: a finite sequence of five terms.

Familiar sequences, and how each one grows:

• natural numbers 1, 2, 3, …: +1 each time;
• odd numbers 1, 3, 5, …: +2 each time;
• triangular numbers 1, 3, 6, 10, 15, …: +2, +3, +4, …;
• square numbers 1, 4, 9, 16, …: +3, +5, +7, ….

Pictures explain them.

• Each triangular number is a sum of natural numbers: 15 = 1 + 2 + 3 + 4 + 5, a triangle of dots with 5 rows.
• Each square number is a sum of odd numbers: 16 = 1 + 3 + 5 + 7. Each bigger square adds an L-shaped border of the next odd number.

Naming terms by position. We write t₁ for the first term, t₂ for the second, and tₙ for the term in position n. In the odd numbers, t₄ = 7. The little number is the position, not the value.

Terms can be fractions or negative: 1, 1/2, 1/3, 1/4, … gets smaller; –7, –3, 1, 5, 9, … goes up by 4.

Predict, then check. Before working out the next term, guess it from the pattern, then test your guess.