Sequences

Maths · Algebra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A sequence is a list of numbers that follows a rule. A term-to-term rule tells you how to get from one term to the next. A position-to-term rule, the nth term, gives any term straight from its position n, so you can find the 100th term without listing the others.

In an arithmetic (linear) sequence you add the same number each time. In a geometric sequence you multiply by the same number each time. In a Fibonacci-type sequence you add the two terms before.

In a quadratic sequence the first differences change but the second differences are constant, and the nth term contains n2.

Rules to remember

Common mistake

For 5, 9, 13, … students write n + 4 because the terms go up in 4s. Going up in 4s means 4n; then adjust to fit the first term, giving 4n + 1.

Worked example

Find the nth term of the quadratic sequence 2, 9, 22, 41, 66, …

  1. First differences: 7, 13, 19, 25. Second differences: 6, 6, 6, so a = 6 ÷ 2 = 3.
  2. Subtract 3n2 (3, 12, 27, 48, 75) from the terms: −1, −3, −5, −7, −9.
  3. This is a linear sequence going down in 2s with first term −1, so its nth term is −2n + 1.
  4. Check with n = 4: 3(16) − 8 + 1 = 41.

Answer: 3n2 − 2n + 1

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