SequencesCambridge IGCSE Maths: Revision notes

A sequence is an ordered list of numbers called terms, each generated by a rule.

• The term-to-term rule describes how to get from one term to the next (e.g. "add 4", "multiply by 2")
• The position-to-term rule (or nth term) gives any term directly from its position $n$, without working through every term before it
• Sequences can be linear, quadratic, cubic, or exponential, depending on how the terms grow

To continue a sequence, first find the term-to-term rule by looking at the differences (or ratios) between consecutive terms.

In a linear sequence, the difference between consecutive terms is constant. This constant is the common difference, $d$.

For a linear sequence with first term $a$ and common difference $d$: $T_n = a + (n-1)d = dn + (a-d)$

In practice, the nth term of a linear sequence always has the form $T_n = dn + c$, where:

• $d$ = common difference (coefficient of $n$)
• $c$ = the value you'd get at position $n=0$ (found by subtracting $d$ from the first term)

Example: For $3, 7, 11, 15, \ldots$, the common difference is $4$, so $T_n = 4n + c$. Since $T_1 = 3$, $c = 3 - 4 = -1$, giving $T_n = 4n - 1$.

In a quadratic sequence, the first differences are not constant, but the second differences (the differences between the first differences) are constant.

The nth term has the form $T_n = an^2 + bn + c$, where:

1. $a = \dfrac{\text{second difference}}{2}$
2. Subtract $an^2$ from each term to leave a linear sequence in $n$, then find that linear sequence's nth term as normal to get $bn + c$

Example: For $2, 5, 10, 17, 26, \ldots$: first differences are $3, 5, 7, 9$; second difference is $2$, so $a = 1$. Subtracting $n^2$ from each term ($2-1, 5-4, 10-9, 17-16, \ldots$) gives $1, 1, 1, 1, \ldots$, so $T_n = n^2 + 1$.

A cubic sequence has an nth term of the form $T_n = an^3 + bn^2 + cn + d$. It is identified because the third differences are constant, while the first and second differences both vary.

To find a cubic nth term:

1. Find first, second and third differences
2. $a = \dfrac{\text{third difference}}{6}$
3. Subtract $an^3$ from the sequence to leave a quadratic sequence, then find its nth term as above

Cubic sequences grow much faster than linear or quadratic sequences as $n$ increases.

Beyond polynomial sequences, exam questions may use:

• Exponential sequences, where each term is found by multiplying the previous term by a constant ratio, e.g. $2, 6, 18, 54, \ldots$ (ratio $3$), with $T_n = a \times r^{n-1}$
• Subscript notation, e.g. $T_n$ or $T_1, T_2, T_3$ to refer to the 1st, 2nd, 3rd terms

When continuing a pattern-based sequence (e.g. matchstick or dot patterns described in words), count how many are added at each stage to find the term-to-term rule before generalising to an nth term.