Sequences

Sequences and term-to-term rules

A sequence is an ordered list of numbers, called terms, that follow a rule. The term-to-term rule tells you how to get from one term to the next:

• $5,8,11,14,\dots$: add 3 each time;
• $3,6,12,24,\dots$: multiply by 2 each time;
• $1,1,2,3,5,8,\dots$: add the two previous terms.

To continue a sequence, find the rule and keep applying it. If the gaps between terms are not equal, look at the gaps themselves. In the syllabus example $1,3,6,10,15,\dots$ the gaps are $2,3,4,5$, growing by 1 each time, so the next two gaps are 6 and 7 and the next two terms are 21 and 28. These are the triangular numbers.

Sequences also come from patterns of shapes. Count the tiles or sticks in the first few patterns, write the counts as a sequence, and look at how each pattern grows from the one before.

Recognise these sequences on sight: square numbers $1,4,9,16,25,\dots$, cube numbers $1,8,27,64,125,\dots$, triangular numbers $1,3,6,10,15,\dots$ and powers of 2, $2,4,8,16,\dots$

The nth term of a linear sequence

A term-to-term rule is slow for finding, say, the 100th term. The $n$th term is a formula that gives any term straight from its position $n$.

In a linear sequence the difference between consecutive terms is constant. Take $7,11,15,19,\dots$, which goes up by 4 each time, and compare it with the 4 times table in the table below. Each term is 3 more than $4n$, so the $n$th term is $4n+3$.

Why this works: going up by $d$ every step is exactly how the $d$ times table behaves, so the $n$th term is $dn$ plus a fixed adjustment. A quick way to find the adjustment is the zero term, the term before the first: for $7,11,15,\dots$ it is $7-4=3$. If the sequence goes down, $d$ is negative: $20,17,14,11,\dots$ has $d=-3$ and zero term 23, so its $n$th term is $23-3n$.

Using the $n$th term. Substitute to find a term: the 50th term of $4n+3$ is $4(50)+3=203$. To test whether a number is in the sequence, set the formula equal to it and solve. $4n+3=99$ gives $n=24$, a whole number, so 99 is the 24th term. $4n+3=100$ gives $n=24.25$, which is not a whole number, so 100 is not in the sequence.

Simple quadratic and cubic sequences

When the differences are not constant, find the second differences, the differences between the differences. If the second differences are constant, the sequence is quadratic and its $n$th term contains $n^2$.

At Core level the quadratic and cubic sequences are simple ones, closely related to $n^2$ or $n^3$, so compare the sequence with the square or cube numbers term by term.

The syllabus example $2,5,10,17,\dots$ has differences $3,5,7$ and second differences 2. Compare it with $n^2$: $1,4,9,16$. Each term is 1 more, so the $n$th term is $n^2+1$.

A second difference of 2 goes with $n^2$, and a second difference of 4 goes with $2n^2$: the number in front of $n^2$ is always half the second difference. So $3,12,27,48,\dots$, with second difference 6, is $3n^2$.

For a cubic sequence the third differences are constant. Compare with the cubes: $0,7,26,63,\dots$ is each cube $1,8,27,64$ minus 1, so its $n$th term is $n^3-1$.

Relationships between sequences

Sequences are often related to each other, and spotting the relationship gives the $n$th term quickly. If you know the $n$th terms of two sequences, you can add, subtract or multiply them term by term.

• $3,8,15,24,\dots$ is $1,4,9,16$ plus $2,4,6,8$, so its $n$th term is $n^2+2n$.
• $2,6,12,20,\dots$ is $1\times 2$, $2\times 3$, $3\times 4$, $4\times 5$, so its $n$th term is $n(n+1)$.
• The triangular numbers are half of these: $\frac{n(n+1)}{2}$ gives $1,3,6,10,\dots$

When a question gives you two sequences and asks about a third made from them, work with the $n$th terms rather than with long lists of numbers.