Patterns — Mathslive.ie

Section 1 of 4

Linear Patterns

A linear pattern grows by the same amount each time (a constant first difference). Its rule is $y = mx + c$: $m$ is the difference, $c$ the starting value.

Time	0	1	2	3
Height	4	6	8	10

Worked example — find the rule from a table

Find the rule for the table above, and show it on a graph.

First difference is $+2$ each time, so $m = 2$

At time $0$ the height is $4$, so $c = 4$

$y = 2x + 4$

Section 2 of 4

Sequences & the $n$th Term

A sequence is a list of terms $T_1, T_2, T_3, \dots$ The rule $T_n$ (the $n$th term) lets you jump to any term.

Worked example — the nth term of a sequence

For $2, 4, 6, 8, \dots$ find $T_n$ and hence $T_{50}$.

Each term is $2\times$ its position, so $T_n = 2n$

$T_{50} = 2(50)$

$T_n = 2n$, $T_{50} = 100$

Section 3 of 4

Spotting a Quadratic

First & second differences

If the first differences are constant, it is linear. If the second differences are constant, it is quadratic: $T_n = an^2 + bn + c$ with $a = \dfrac{\text{second difference}}{2}$.

Now you try

You try

For $T_n = n^2 + 2n + 3$, list $T_0$ to $T_4$ and show the pattern is quadratic.

Pen and paper out — try it before you reveal.

$T_0=3,\ T_1=6,\ T_2=11,\ T_3=18,\ T_4=27$

First differences: $3, 5, 7, 9$

Second differences: $2, 2, 2$ — constant

constant second difference $\Rightarrow$ quadratic

Section 4 of 4

Finding the Quadratic Rule

You try

Find $T_n$ for the pattern $1, 6, 13, 22, 33, \dots$

Pen and paper out — try it before you reveal.

First differences $5, 7, 9, 11$; second differences $2, 2, 2$

$a = \dfrac{2}{2} = 1$, so $T_n = n^2 + bn + c$

$T_1 = 1$: $1 + b + c = 1 \Rightarrow b + c = 0$

$T_2 = 6$: $4 + 2b + c = 6 \Rightarrow 2b + c = 2$

Subtract: $b = 2$, then $c = -2$

$T_n = n^2 + 2n - 2$ (check $T_3 = 9+6-2 = 13$ ✓)

More examples

You try

Find $T_n$ for the sequence $1, 2, 3, 4, \dots$

How does each term relate to its position?

Each term equals its position

$T_n = n$

You try

Find $T_n$ for the sequence $2, 4, 6, 8, \dots$

Twice the position each time.

$T_n = 2\times n$

$T_n = 2n$

That’s Patterns.

Linear rules from tables, the $n$th term, spotting a quadratic from second differences, and building $T_n = an^2+bn+c$.