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4.6 Chapter summary

  • A list of numbers that form a pattern is called a sequence. Each number in a sequence is called a term of the sequence. The first number is the first term of the sequence.
  • Numbers that follow one another are said to be consecutive.
  • When the differences between consecutive terms of a sequence are the same, we say the difference is constant, or that the sequence has a common difference. The number pattern \(4\); \(7\); \(10\); \(13\); \(16\) has a common difference of \(3\).
  • If we multiply or divide by a number to get the next term in the sequence, then the ratio between the first and second numbers is the same as the ratio between the second and third numbers.
  • Sequences in which the number we multiply or divide by remains the same therefore have a constant ratio.
  • There are a few special non-linear patterns that you should know and recognise:

    Name List Formula
    Perfect squares \(1; 4; 9; 16;\ldots\) \(t = n^{2}\)
    Perfect cubes \(1; 8; 27; 64;\ldots\) \(t = n^{3}\)
    Powers of \(2^{2}\) \(2; 4; 8; 16;\ldots\) \(t = 2^{n}\)
    Powers of \(3^{3}\) \(3; 9; 27; 81;\ldots\) \(t = 3^{n}\)
    Reciprocals \(1; 1^{2};1^{3};1^{4};\ldots\) \(t = 1^{n}\)
  • Given a sequence of numbers, we can identify a pattern or relationship between the term and its position in the sequence. This allows us to predict a term in a sequence based on the position of that term in the sequence.
  • It is useful to represent sequences in tables so that we can easily see the position of the term. It also makes it easier to describe the general rule for the pattern. For example, if the rule is: “Multiply the position of the number by \(2\) and subtract \(1\) from the answer.” We can write this rule as a number sentence: \(\text{position of the number} \times 2 - 1\)
  • Geometric patterns are number patterns represented diagrammatically. The diagrammatic representation reveals the structure of the number pattern.
  • Linear patterns always have a formula that includes the term number multiplied by the constant difference, as well as a constant term.