Sequences

What it is

A sequence is numbers written in a fixed order: a₁, a₂, a₃, … Each number is a term, and the index n gives the position. A sequence can be finite, with a last term, or infinite.

There are two ways to give a sequence:

• An explicit formula, where the term at position n is worked out directly. For aₙ = 2n, a₅ = 10 and a₁₀₀ = 200, without going through the terms before.
• Recursion, where each term is worked out from the one or ones before it, together with a starting term. With a₁ = 3 and aₙ = 2 · aₙ₋₁ − 1 the sequence is 3, 5, 9, 17, 33, …

Examples:

• 2, 4, 6, 8, … with aₙ = 2n. Each term is 2 more than the last.
• 7, 10, 13, … with aₙ = 3n + 4. See arithmetic sequence.
• 5, 10, 20, 40, … with aₙ = 5 · 2ⁿ⁻¹. See geometric sequence.
• 1, 1, 2, 3, 5, 8, … (the Fibonacci sequence), where each term is the sum of the two before: 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8. The tenth term is 55.
• 1, 4, 9, 16, … with aₙ = n². The differences 3, 5, 7 grow and the ratios 4 and 2.25 vary, so it is neither arithmetic nor geometric.

These are not sequences:

• The set {2, 4, 6, 8}. In a set the order does not matter; in a sequence it does: 2, 4, 6, 8 and 8, 6, 4, 2 are different sequences.
• The sum 2 + 4 + 6 + 8 = 20. That is a single number. The sum of a sequence’s terms is called a series, see sum of an arithmetic series.

Why it matters

A sequence is a function whose inputs are the positions 1, 2, 3, …, which is why it is drawn as separate points rather than a connected curve. The idea turns up wherever something repeats step by step: savings that grow with deposits, a dose that builds up and wears off, month after month.

• Equal steps give an arithmetic sequence, see also the common difference.
• Equal factors give a geometric sequence.
• Both are special cases of the recursive sequence, aₙ = k · aₙ₋₁ + m, which works out terms one at a time.

Common misconceptions

• “Every sequence is arithmetic or geometric.” The square numbers 1, 4, 9, 16 are neither.
• “Three terms settle the pattern.” 1, 2, 4 can continue with 8 (doubling) or with 7 (steps 1, 2, 3 growing). Both sequences start 1, 2, 4. A pattern needs a rule, not just a few terms.
• “The recursion formula is enough.” Without the starting term the rule aₙ = 2aₙ₋₁ − 1 fits several sequences: start at 3 and you get 3, 5, 9, …; start at 1 and you get 1, 1, 1, …
• “The index always starts at 1.” Many books start at 1, others at 0; that changes what “the tenth term” is. Check what the problem assumes.

Common questions