Sequences & Functions

Describing Sequences

A sequence is an ordered list of numbers called terms. We label them $T_1, T_2, T_3, \dots$ where $T_n$ is the term in position $n$.

There are two ways to describe how a sequence is built.

A term-to-term rule tells you how to get the next term from the previous one. For $3, 7, 11, 15, \dots$ the rule is "add $4$".
A position-to-term rule (the nth term) gives any term directly from its position $n$. For the same sequence, $T_n = 4n - 1$, so $T_{50} = 4(50) - 1 = 199$.

Key terms

Term — a single number in a sequence.

nth term — a formula in $n$ that generates any term from its position.

Term-to-term rule — how each term relates to the one before it.

The position-to-term rule is far more powerful: it lets you jump straight to the $100$th term without listing everything in between.

The nth Term of a Linear Sequence

A linear (or arithmetic) sequence goes up or down by the same amount each time. This constant step is the common difference $d$.

The nth term has the form:

$$T_n = dn + c$$

To find it:

1. Find the common difference $d$ (the first differences).
2. The coefficient of $n$ is $d$.
3. Find $c$ by working back: $c = T_1 - d$.

Worked example

Find the nth term of $5, 8, 11, 14, \dots$ and use it to find the $20$th term.

The differences are all $+3$, so $d = 3$ and the rule starts $T_n = 3n + c$.

Exam tip

A quick way to spot the constant: the coefficient of $n$ always equals the common difference. If a sequence decreases, $d$ is negative, e.g. $20, 17, 14, \dots$ gives $T_n = -3n + 23$.

The nth Term of a Quadratic Sequence

A quadratic sequence has a constant second difference (the differences of the differences). Its nth term has the form:

$$T_n = an^2 + bn + c$$

The method:

1. Find the second difference. Then $a = \dfrac{\text{second difference}}{2}$.
2. Subtract $an^2$ from each term to leave a linear sequence.
3. Find the nth term of that linear part to get $bn + c$.

Worked example

Find the nth term of $4, 7, 12, 19, 28, \dots$

First differences: $3, 5, 7, 9$. Second differences: $2, 2, 2$ (constant), so this is quadratic.

$a = \dfrac{2}{2} = 1$, so the $n^2$ part is $1n^2$.

Watch out

Always halve the second difference, not the first, to find $a$. A common slip is forgetting that the linear part can itself have a non-zero coefficient of $n$ — always subtract $an^2$ and treat what's left as a new sequence.

Special Sequences

You should recognise these on sight.

A geometric sequence multiplies by a fixed common ratio $r$ each time. For $2, 6, 18, 54, \dots$ the ratio is $r = 3$, so $T_n = 2 \times 3^{\,n-1}$.

Real world

Geometric sequences model anything that grows by a fixed percentage: money in a savings account, a population, or the spread of a rumour. Fibonacci numbers appear in the spirals of sunflowers and pinecones.

Function Notation and Evaluating Functions

A function is a rule that takes an input and gives exactly one output. We write $f(x)$, read "f of x", where $x$ is the input.

For example, if $f(x) = 3x - 2$, then to evaluate at a value we substitute it for $x$:

$f(4) = 3(4) - 2 = 10$
$f(-1) = 3(-1) - 2 = -5$

You can also solve equations involving functions. If $f(x) = 17$, then $3x - 2 = 17$, so $x = \dfrac{19}{3}$.

Key terms

Function — a rule giving one output for each input.

Argument — the input value, the thing inside the brackets.

Evaluate — substitute a number and simplify.

Composite Functions

A composite function applies one function and then feeds the result into another. The notation $fg(x)$ means "do $g$ first, then $f$":

$$fg(x) = f\big(g(x)\big)$$

Think of two machines in a row — the output of the first becomes the input of the second.

Worked example

Given $f(x) = 2x + 1$ and $g(x) = x^2$, find $fg(3)$ and an expression for $fg(x)$.

Watch out

$fg(x)$ is not $f(x) \times g(x)$. It means substitute $g(x)$ into $f$. And in general $fg(x) \neq gf(x)$ — always apply the function nearest to $x$ first.

Inverse Functions

The inverse function $f^{-1}(x)$ reverses what $f$ does: if $f$ takes $3$ to $10$, then $f^{-1}$ takes $10$ back to $3$. It "undoes" the original machine.

To find $f^{-1}(x)$:

1. Write $y = f(x)$.
2. Rearrange to make $x$ the subject.
3. Swap $y$ for $x$ — this gives $f^{-1}(x)$.

Worked example

Find the inverse of $f(x) = \dfrac{x + 4}{3}$.

Write $y = \dfrac{x+4}{3}$.

Exam tip

$f^{-1}(x)$ does not mean $\dfrac{1}{f(x)}$. The $-1$ is notation for "inverse", not a power. A neat check: $f f^{-1}(x) = x$ and $f^{-1} f(x) = x$ — applying a function then its inverse always returns the original input.