Unit 9 · Sequences and functions

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Every rule, key word and worked example from all six sections. Use your browser's print command to put it on paper, or download the PDF.

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9.1 Generating sequences Algebra

  • A sequence is an ordered list; each number in it is a term.
  • A term-to-term rule says how to get from one term to the next.
  • To generate a sequence you need the rule and the first term — one without the other is not enough.
  • A sequence is linear when the gap between terms is always the same. That gap is the common difference.
  • To find the common difference, subtract any term from the one after it — and check the gap really is constant.
  • To reach the kth term from the first, take k − 1 steps, not k. That off-by-one is the commonest slip.
  • Not every sequence is linear: 2, 6, 18, 54 multiplies by 3; 1, 4, 9, 16 squares the position.
  • Learn the square, cube and triangular numbers by sight — they come up all year.

Key words: sequence  ·  term  ·  term-to-term rule  ·  first term  ·  common difference  ·  linear sequence  ·  square numbers  ·  triangular numbers

3, 7, 11, 15, 19, …
1st term 3 2nd term 7 3rd term 11
first term 3, rule “add 4”
3 → 7 → 11 → 15 → 19
rule “add 4”, first term 1 → 1, 5, 9, 13
rule “add 4”, first term 3 → 3, 7, 11, 15
5, 9, 13, 17
9 − 5 = 4 13 − 9 = 4 17 − 13 = 4

9.2 Using the nth term Algebra

  • The nth term rule gives any term straight from its position, with no stepping.
  • n = 1 for the first term, n = 2 for the second, and so on.
  • The coefficient of n is the common difference. A gap of 4 means the rule starts 4n.
  • Find the constant by comparing 4n with the sequence: what must be added to reach it?
  • For a decreasing sequence the coefficient is negative: 20, 17, 14 has rule −3n + 23.
  • Always check by substituting n = 1, 2, 3. It takes seconds and catches almost everything.
  • To find a term's position, set the rule equal to it and solve.
  • If solving gives a fraction, that number is not in the sequence — positions must be whole numbers.

Key words: nth term  ·  position  ·  coefficient  ·  substitute  ·  position-to-term rule

term-to-term: 3 → 7 → 11 → 15 → … 99 steps
nth term 4n − 1: put n = 100 → 399
5, 9, 13, 17 gap 4 → starts 4n
20, 17, 14 gap −3 → starts −3n
want 7, 10, 13, 16
3n gives 3, 6, 9, 12
always 4 more → 3n + 4
rule 3n + 4

9.3 Functions and function machines Algebra

  • A function turns each input into exactly one output.
  • A function machine shows the operations in order, left to right.
  • To go forwards, apply the operations in order.
  • To go backwards, use the inverse of each operation, in reverse order.
  • Inverses: + and −; × and ÷.
  • Order matters. “× 2 then + 1” is y = 2x + 1; “+ 1 then × 2” is y = 2(x + 1).
  • A mapping diagram draws an arrow from each input to its output.
  • Writing the machine as y = … turns it into algebra, which is where Unit 11 picks it up.

Key words: function  ·  function machine  ·  input  ·  output  ·  inverse  ·  mapping diagram

input 4 → [ × 3 ] → output 12
input 7 → [ × 3 ] → output 21
input 5 → [ × 2 ] → 10 → [ + 1 ] → 11
input 5 → [ + 1 ] → 6 → [ × 2 ] → 12
different order, different answer
output 25, machine × 3 then + 4
undo + 4: 25 − 4 = 21
undo × 3: 21 ÷ 3 = 7