Other Sequences (Cambridge (CIE) O Level Maths): Revision Note

Types of Sequences

What are common types of sequences?

• Sequences can follow any rule, but common sequences are

  • Linear

    • nth term = $an+b$

  • Quadratic

    • nth term = $a{n}^2+bn+c$

  • Cubic

    • nth term = $a{n}^3+b{n}^2+cn+d$

  • Exponential (Geometric)

    • nth term = $a\times r^n$

• Sequences may also be formed using common numbers

  • Prime numbers

    • 2, 3, 5, 7, 11, ...

  • Triangular numbers

    • 1, 3, 6, 10, 15, ...

What is a cubic sequence?

• A cubic sequence has an n th term formula that involves n³

• The third differences are constant (the same)

  • These are the differences between the second differences

  • For example,   4, 25, 82, 193, 376, 649, ...
    1st Differences:  21, 57, 111, 183, 273, ...

    2nd Differences:   36,  54,  72,   90, ...
    3rd Differences:      18,    18,   18, ...

How do I find the n^th term formula for a simple cubic sequence?

• The sequence with the n th term formula of n³ is the cube numbers 

  • 1, 8, 27, 64, 125, ...

    • From 1³, 2³, 3³, 4³, ...

• Finding the n th term formula of other cubic sequences comes from comparing them to the cube numbers, n³

  • 2, 9, 28, 65, 126, ... has the formula n³ + 1

    • Each term is one more than the cube numbers

  • 2, 16, 54, 128, 250, ...  has the formula 2n³

    • Each term is double a cube number

  • 8, 27, 64, 125, ... has the formula (n+1)³

    • They are the cube numbers starting from 2³

• You can also use third differences to help find the n th term $a{n}^3+b{n}^2+cn+d$

  • The value of a  is $\frac{1}{6}$ of the third difference

    • e.g. the third difference of 8, 23, 62, 137, 260, ... is 12

  • You can then subtract an³ from the sequence and find the nth term of the differences

    • e.g. (8, 23, 62, 137, 260, ...) - (2, 16, 54, 128, 250, ... ) is (6, 7, 8, 9, 10, ...) which has nth term n + 5

    • So the nth term is 2n³ + n + 5

What is an exponential (geometric) sequence? 

• An exponential (geometric) sequence is one where you multiply each term by the same number to get the next term

  • E.g. 3, 6, 12, 24, 48, ... is exponential because:

    • terms are multiplied by 2 each time

    • 2 is called the common ratio (or constant multiplier)

    • You can find this by dividing any term by the term immediately before, 6 ÷ 3 or 12 ÷ 6 or 24 ÷ 12 etc

How do I find the n^th term formula for a simple exponential sequence?

• The sequence with the n th term formula of rⁿ is the powers of r

  • e.g. 2, 4, 8, 16, 32, ... has formula 2ⁿ

• Finding the n th term formula of other exponential sequences comes from comparing them to the powers of the multiplier, rⁿ

  • 6, 12, 24, 48, 96, ... has the formula 3× 2ⁿ

    • Each term is three times more than a power of 2

  • 4, 8, 16, 32, 64, ... has the formula 2ⁿ⁺¹

    • Each term is a power of 2 starting at 2²

• If the common ratio satisfies

  • $r>1$ then the sequence increases

  • $0<r<1$ then the sequence decreases

    • E.g. $r=\frac{1}{2}$

How can sequences be made harder?

• You may be given a fraction with two different sequences on the top and bottom

  • E.g. $\frac{3}{1},\frac{5}{8},\frac{7}{27},\frac{9}{64},...$

    • The numerators are the linear sequence $2n+1$

    • The denominators are the cube numbers, $n^3$

    • So the n th term formula is $\frac{2n+1}{n^3}$

• You may be asked to find combinations of two different sequences

  • E.g. if sequence U is the prime numbers and sequence V has the n th term formula $4n^2$, find the sequence U + V

    • U = 2, 3, 5, 7, ... and V = 4, 16, 36, 64, ...

    • U + V = (2 + 4), (3 + 16), (5 + 36), (7 + 64), ... = 6, 19, 41, 71, ...

• Other problems involving setting up and solving equations

  • This may lead to a pair of simultaneous equations