Continuing Sequences (Cambridge (CIE) IGCSE International Maths)

Continuing Sequences

How do I continue a given sequence? 

• You can work out the first differences to see if there is a pattern

  • The first differences are the values the sequence changes by each time

• For example

  • 

    • The first differences are all +3

    • The next term is 13 + 3 = 16

  • 

    • The first differences are all -5

    • The next term is -9 - 5 = -14 

  • 

    • The first differences increase by 1

    • The next term is 23 + 9 = 32

  •  

    • The first differences double each time

    • The next term is 15 + 16 = 31

Sequences of squares, cubes, and triangular numbers

• Sequences can often be formed using square, cube, or triangular numbers

• It can help to be familiar with these sequences of numbers

• Square numbers are the results of squaring integers

  • 1²,  2²,  3²,  4²,  5²,  6²,  7²,  8²,  9²,  10²,  11²,  12²,  ...

  • 1,  4,  9,  16,  25,  36,  49,  64,  81,  100,  121,  144,  ...

• Cube numbers are the results of cubing integers

  • 1³,  2³,  3³,  4³,  5³,  ...

  • 1,  8,  27,  64,  125,  ...

• Triangular numbers are the result of summing consecutive integers

  • 1,  1+2,  1+2+3,  1+2+3+4,  1+2+3+4+5,  ...

  • 1,  3,  6,  10,  15, ...

  • When drawn as dots, triangular numbers form triangles

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