Sequences

Terms and positions

• A sequence is a list of numbers written in a fixed order and governed by a rule
• Each number in the list is a term, and where it sits in the list is its position
• The letter n stands for the position, so n = 1 marks the first term, n = 2 the second, and so on
• Writing the position above each term before you start is the single most useful habit in this topic
• A sequence can also be described by a growing pattern of shapes, in which case counting the dots, sticks or squares in each picture produces the numbers

Two ways of describing a rule

• A term-to-term rule says what to do to one term to reach the next, such as add 9 each time or halve each time
  • It is quick to use but only moves one step at a time, so reaching the 60th term means writing out 59 steps
• A position-to-term rule, usually called the nth term, is an expression in n that produces any term directly
  • With nth term 6n − 1, substituting n = 1, 2, 3 gives 5, 11, 17, and substituting n = 100 gives 599 in one line
  • With nth term n² − 2, the first three terms are −1, 2 and 7, and the 100th term is 9998
• A question asking for a specific far-off term is really asking you to find the nth term first

Subscript notation

• A sequence can be named with a letter, and its terms labelled by position using a subscript
• If a sequence is called T, then T₁ is its first term, T₅ its fifth, and Tₙ its nth
• Questions use this to link two sequences without writing either out, for example by defining a third sequence as Cₙ = Aₙ + Bₙ
• The subscript is a position label, never a multiplier