Sec 1 · G3 (Express)

Number sequences and patterns

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1. Common sequences

Sequence	Terms	Rule
odd numbers	$1, 3, 5, 7, \ldots$	$2n - 1$
square numbers	$1, 4, 9, 16, \ldots$	$n^{2}$
cube numbers	$1, 8, 27, 64, \ldots$	$n^{3}$
triangular numbers	$1, 3, 6, 10, \ldots$	add $2$ , $3$ , $4$ , …

📖 Worked example

Write down the next term of the sequence $-10, \ -15, \ -20, \ -25, \ \ldots$ .

Step 1 of 2

Find the common difference: a term minus the one before it

\mathstrut -15 - (-10) =

-5

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2. The general term

Adds 4 each time: start with 4n

\displaystyle 5, 9, 13, 17, \ldots

$4n$ gives $4, 8, 12, 16$ : one less
$T_n = 4n + 1$

Goes down by 3: start with −3n

\displaystyle 20, 17, 14, 11, \ldots

$-3n$ gives $-3, -6, -9, -12$ : $23$ more
$T_n = 23 - 3n$

Use the general term: the 50th term of 4n + 1

\displaystyle T_{50} = 4(50) + 1

e.g.

\displaystyle 4(50) + 1

\displaystyle = 201

Is 150 a term of 4n + 1?

\displaystyle 4n + 1 = 150

e.g.

\displaystyle 4n + 1 = 150

\displaystyle n = \dfrac{149}{4}

$n$ is not a whole number, so $150$ is not a term

📖 Worked example

Find an expression, in terms of $n$ , for the $n$ th term of the sequence $-6, \ -4, \ -2, \ 0, \ \ldots$ .

Step 1 of 3

Find the common difference: it is the number in front of n

\mathstrut (-4) - (-6) =

2

📖 Worked example

Find the 25th term of the sequence $-5, \ -3, \ -1, \ 1, \ \ldots$ .

Step 1 of 4

Find the common difference: it is the number in front of n

\mathstrut (-3) - (-5) =

2

Sequence	Compare with	nth term
$3, 6, 11, 18, \ldots$	$1, 4, 9, 16$ (subtract: always $2$ )	$n^{2} + 2$
$2, 6, 12, 20, \ldots$	$1, 4, 9, 16$ (subtract: $1, 2, 3, 4$ )	$n^{2} + n$
$\dfrac{1}{3}, \dfrac{3}{5}, \dfrac{5}{7}, \ldots$	tops $2n - 1$ , bottoms $2n + 1$	$\dfrac{2n - 1}{2n + 1}$

📖 Worked example

Find an expression, in terms of $n$ , for the $n$ th term of the sequence $\dfrac{1}{3}, \ \dfrac{3}{5}, \ \dfrac{5}{7}, \ \dfrac{7}{9}, \ \ldots$ .

Step 1 of 3

The numerators go up by the same amount each time: find their nth term

2n - 1

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3. Patterns

Count, find the difference, then the general term

\displaystyle 4, 7, 10, 13, \ldots

Matchstick squares in a row: Figure 1 has 4 sticks, Figure 2 has 7, Figure 3 has 10, Figure 4 has 13.

each new square adds $3$ sticks
$T_n = 3n + 1$

📖 Worked example

The diagram shows Figures 2, 3 and 4 of a pattern made with matchsticks. Find an expression, in terms of $n$ , for the number of matchsticks in Figure $n$ .

Step 1 of 4

Count the matchsticks in each figure (a side shared by two shapes is one matchstick)

\text{Figure 2:}

11

\text{Figure 3:}

16

\text{Figure 4:}

21

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Practice questions by skill

1. Common sequences

Sequence	Terms	Rule
odd numbers	$1, 3, 5, 7, \ldots$	$2n - 1$
square numbers	$1, 4, 9, 16, \ldots$	$n^{2}$
cube numbers	$1, 8, 27, 64, \ldots$	$n^{3}$
triangular numbers	$1, 3, 6, 10, \ldots$	add $2$ , $3$ , $4$ , …

📖 Worked example

Write down the next term of the sequence $-10, \ -15, \ -20, \ -25, \ \ldots$ .

Step 1 of 2

Find the common difference: a term minus the one before it

\mathstrut -15 - (-10) =

-5

Practise this →

2. The general term

Adds 4 each time: start with 4n

\displaystyle 5, 9, 13, 17, \ldots

$4n$ gives $4, 8, 12, 16$ : one less
$T_n = 4n + 1$

Goes down by 3: start with −3n

\displaystyle 20, 17, 14, 11, \ldots

$-3n$ gives $-3, -6, -9, -12$ : $23$ more
$T_n = 23 - 3n$

Use the general term: the 50th term of 4n + 1

\displaystyle T_{50} = 4(50) + 1

e.g.

\displaystyle 4(50) + 1

\displaystyle = 201

Is 150 a term of 4n + 1?

\displaystyle 4n + 1 = 150

e.g.

\displaystyle 4n + 1 = 150

\displaystyle n = \dfrac{149}{4}

$n$ is not a whole number, so $150$ is not a term

📖 Worked example

Find an expression, in terms of $n$ , for the $n$ th term of the sequence $-6, \ -4, \ -2, \ 0, \ \ldots$ .

Step 1 of 3

Find the common difference: it is the number in front of n

\mathstrut (-4) - (-6) =

2

📖 Worked example

Find the 25th term of the sequence $-5, \ -3, \ -1, \ 1, \ \ldots$ .

Step 1 of 4

Find the common difference: it is the number in front of n

\mathstrut (-3) - (-5) =

2

Sequence	Compare with	nth term
$3, 6, 11, 18, \ldots$	$1, 4, 9, 16$ (subtract: always $2$ )	$n^{2} + 2$
$2, 6, 12, 20, \ldots$	$1, 4, 9, 16$ (subtract: $1, 2, 3, 4$ )	$n^{2} + n$
$\dfrac{1}{3}, \dfrac{3}{5}, \dfrac{5}{7}, \ldots$	tops $2n - 1$ , bottoms $2n + 1$	$\dfrac{2n - 1}{2n + 1}$

📖 Worked example

Find an expression, in terms of $n$ , for the $n$ th term of the sequence $\dfrac{1}{3}, \ \dfrac{3}{5}, \ \dfrac{5}{7}, \ \dfrac{7}{9}, \ \ldots$ .

Step 1 of 3

The numerators go up by the same amount each time: find their nth term

2n - 1

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3. Patterns

Count, find the difference, then the general term

\displaystyle 4, 7, 10, 13, \ldots

each new square adds $3$ sticks
$T_n = 3n + 1$

📖 Worked example

The diagram shows Figures 2, 3 and 4 of a pattern made with matchsticks. Find an expression, in terms of $n$ , for the number of matchsticks in Figure $n$ .

Step 1 of 4

Count the matchsticks in each figure (a side shared by two shapes is one matchstick)

\text{Figure 2:}

11

\text{Figure 3:}

16

\text{Figure 4:}

21

Practise this →