Continuing Sequences (Cambridge International IGCSE Mathematics)

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

  • 4 comma space space space space space 7 comma space space space space space space 10 comma space space space space space 13 $( + 3 ) ( + 3 ) ( + 3 )$

    • The first differences are all +3
    • The next term is 13 + 3 = 16

  • 6 comma space space space space space 1 comma space space space space space minus 4 comma space space space space minus 9 $( - 5 ) ( - 5 ) ( - 5 )$

    • The first differences are all -5
    • The next term is -9 - 5 = -14

  • 2 comma space space space space space 8 comma space space space space space space 15 comma space space space space space 23 $( + 6 ) ( + 7 ) ( + 8 )$

    • The first differences increase by 1
    • The next term is 23 + 9 = 32

  • 1 comma space space space space space space 3 comma space space space space space space space 7 comma space space space space space space 15 $( + 2 ) ( + 4 ) ( + 8 )$

    • 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

    • $\circ$ $\circ\,\circ\circ$ $\circ\,\circ\circ\,\circ\circ\circ$

Worked Example Find the next term in the sequence

16, 19, 25, 37, 61

Answer:

Look at the first differences (the values the sequence changes by each time)

16 comma space space space space space space space 19 comma space space space space space space space space 25 comma space space space space space space space space 37 comma space space space space space space space space 61 $> ( + 3 ) ( + 6 ) ( + 12 ) ( + 24 )$

The differences are doubling each time, so the next difference will be 2 × 24 = 48

61 + 48

Final answer: The next term is 109