Tiles in each stage: T=2n+1

01 / LEARN

The idea, step by step

See how the tiles grow

Each stage has one centre tile and two arms. At stage $n$ , each arm contains $n$ tiles. Stage $1$ has $1 + 1 + 1 = 3$ tiles; stage $2$ has $2 + 1 + 2 = 5$ ; stage $3$ has $3 + 1 + 3 = 7$ .

Moving to the next stage adds one tile to each arm, so the total increases by $2$ . Count the centre only once. The structure gives the stage rule $T = 2 n + 1$ , where $T$ is the total number of tiles and $n = 1, 2, 3, \dots$ .

T = 2 n + 1

Stage 1

3 tiles

Stage 2

5 tiles

Stage 3

7 tiles

One centre tile. Each arm grows by one tile at every stage.

Tiles in each stage: T=2n+1

Number of tiles (T)

Stage (n)

Constructed tile pattern; positive whole-number stages only

Make a table and check the rule

Substitute each stage number into the rule: multiply it by $2$ , then add $1$ . For example, stage $4$ has $T = 2 (4) + 1 = 9$ tiles.

The constant difference $2$ tells us how much the total changes between consecutive stages. It does not tell us the starting total by itself: $T = 2 n + 4$ also increases by $2$ . Check a proposed rule against the actual stages, including stage $1$ .

Tile totals for the first four stages of $T=2n+1$
Stage $n$	Calculation	Tiles $T$
1	$2 (1) + 1$	3
2	$2 (2) + 1$	5
3	$2 (3) + 1$	7
4	$2 (4) + 1$	9

Turn the table into coordinates

Put stage number $n$ on the horizontal axis and total tiles $T$ on the vertical axis. Each table row becomes an ordered pair $(n, T)$ , so stage $3$ is the point $(3, 7)$ . Label both axes and use an even scale on each axis.

Plot $(1, 3)$ , $(2, 5)$ , $(3, 7)$ and $(4, 9)$ as separate points. Each move of $1$ stage corresponds to $2$ more tiles. Although the points line up, keep them separate for this pattern: only positive whole-number stages exist, so a point halfway between stages is not another physical stage.

Build the pattern

Stage 1

Stage n1

T = 2 × 1 + 1 = 3 tiles

Each stage becomes a point (`n`, `T`).
Stage `n`	Tiles `T`	Point
1	3	(1, 3)
2	5	(2, 5)
3	7	(3, 7)
4	9	(4, 9)
5	11	(5, 11)
6	13	(6, 13)

A common trap

The pattern starts $3, 5, 7$ , so 'add $2$ ' means its stage rule is $T = n + 2$ .

Instead: 'Add $2$ ' describes moving from one total to the next. A stage rule uses the stage number directly. At stage $2$ , $n + 2 = 4$ , but there are $5$ tiles. The structure gives two arms of $n$ tiles plus one centre: $T = 2 n + 1$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Find a later stage without drawing every tile

The two-arm tile pattern follows $T = 2 n + 1$ . How many tiles are in stage $8$ ? Explain the two parts of the calculation.

Substitute $n = 8$ : $T = 2 (8) + 1$ .
The two arms contain $2 \times 8 = 16$ tiles altogether. Add the single centre tile: $16 + 1 = 17$ .
Check the structure: $8$ tiles on each arm and $1$ centre tile make $8 + 1 + 8 = 17$ .

Stage

8

has

17

tiles.

Example 2

Use a different rule to make a table

For the rule $y = 3 x + 2$ , find $y$ when $x = 0, 1, 2$ and list the corresponding coordinates.

For $x = 0$ , $y = 3 (0) + 2 = 2$ .
For $x = 1$ , $y = 3 (1) + 2 = 5$ . For $x = 2$ , $y = 3 (2) + 2 = 8$ .
Write each input first and its matching output second. The pairs are $(0, 2)$ , $(1, 5)$ and $(2, 8)$ .
When $x$ increases by $1$ , $y$ increases by $3$ , matching the multiplier in the rule.

The

y

-values are

2, 5, 8

and the points are

(0, 2)

(1, 5)

and

(2, 8)

Example 3

Work backwards and check the stage

Which stage of $T = 2 n + 1$ has $19$ tiles? Give its graph coordinates.

Set the total to $19$ : $2 n + 1 = 19$ .
Subtract $1$ from both sides: $2 n = 18$ . Divide both sides by $2$ : $n = 9$ .
Check: $2 (9) + 1 = 19$ . The result is a positive whole-number stage.
The stage is the horizontal coordinate and tiles are the vertical coordinate, so plot $(9, 19)$ .

Stage

9

; the point is

(9, 19)

Check your understanding

For $T = 2 n + 1$ , a student plots $(11, 5)$ to represent stage $5$ . Correct the point and explain your change.

Show answer

Plot $(5, 11)$ .

Stage $5$ has $2 (5) + 1 = 11$ tiles. An ordered pair here is $(n, T)$ : stage first, total tiles second.

03 / WORKSHEET

Your turn

8 questions · 24 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

The pictured pattern has a centre tile and two arms. Stage $1$ has $3$ tiles, stage $2$ has $5$ , and stage $3$ has $7$ . Describe what is added at each step and find the number of tiles in stage $4$ .
2 marks
Use $T = 2 n + 1$ to complete this table: stages $n = 1, 2, 3, 4, 6$ have totals $T = 3, 5, □, □, □$ . Show each missing calculation.
3 marks
Find the total tiles in stage $15$ of $T = 2 n + 1$ . Explain what the $2 n$ and the $+ 1$ represent in the picture.
3 marks
A student proposes $T = n + 2$ for the pattern $3, 5, 7, 9, \dots$ because the total goes up by $2$ . Test the rule at stages $1$ and $2$ , then explain why the constant difference is not a complete stage rule.
3 marks
Draw a Cartesian graph for stages $1$ to $4$ of $T = 2 n + 1$ . Label the horizontal axis 'Stage $n$ ' and the vertical axis 'Tiles $T$ '. Use a scale of $1$ per interval on both axes, plot the four points and explain why they should remain separate points.
4 marks
Which stage of $T = 2 n + 1$ has $25$ tiles? Show your calculation and check your answer by substitution.
3 marks
The horizontal axis is stage number and the vertical axis is number of tiles. Which point represents stage $7$ of $T = 2 n + 1$ : $(7, 15)$ or $(15, 7)$ ? Explain the coordinate order.
2 marks
A different growing pattern has the rule $B = 3 n + 1$ . Make a table for stages $n = 1, 2, 3, 4$ , list the four coordinates $(n, B)$ and state how the total changes when the stage increases by $1$ .
4 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

The pictured pattern has a centre tile and two arms. Stage $1$ has $3$ tiles, stage $2$ has $5$ , and stage $3$ has $7$ . Describe what is added at each step and find the number of tiles in stage $4$ .
2 marks
Each arm grows by one tile, so the total increases by $2$ . (1 mark)
Stage $4$ has $7 + 2 = 9$ tiles. (1 mark)
One tile is added to each arm, making $2$ extra tiles. Stage $4$ has $9$ tiles.
Use $T = 2 n + 1$ to complete this table: stages $n = 1, 2, 3, 4, 6$ have totals $T = 3, 5, □, □, □$ . Show each missing calculation.
3 marks
At $n = 3$ , $T = 2 (3) + 1 = 7$ . (1 mark)
At $n = 4$ , $T = 2 (4) + 1 = 9$ . (1 mark)
At $n = 6$ , $T = 2 (6) + 1 = 13$ . The last column skips stage $5$ , so do not just add $2$ to the previous displayed total. (1 mark)
The missing totals are $7, 9, 13$ .
Find the total tiles in stage $15$ of $T = 2 n + 1$ . Explain what the $2 n$ and the $+ 1$ represent in the picture.
3 marks
$T = 2 (15) + 1 = 31$ tiles. (1 mark)
$2 n$ counts two arms with $n$ tiles in each. (1 mark)
$+ 1$ counts the single centre tile. (1 mark)
$31$ tiles; $2 n$ counts the two arms and $+ 1$ counts the centre tile.
A student proposes $T = n + 2$ for the pattern $3, 5, 7, 9, \dots$ because the total goes up by $2$ . Test the rule at stages $1$ and $2$ , then explain why the constant difference is not a complete stage rule.
3 marks
At $n = 1$ , $n + 2 = 3$ , so the proposed rule matches the first stage. (1 mark)
At $n = 2$ , $n + 2 = 4$ , which does not match the actual total $5$ . (1 mark)
A difference tells how consecutive totals change. A stage rule must connect each stage number to its total; here $T = 2 n + 1$ . (1 mark)
It gives $3$ at stage $1$ but $4$ at stage $2$ , so it fails. The difference describes the change between stages; $T = 2 n + 1$ gives the total directly from the stage number.
Draw a Cartesian graph for stages $1$ to $4$ of $T = 2 n + 1$ . Label the horizontal axis 'Stage $n$ ' and the vertical axis 'Tiles $T$ '. Use a scale of $1$ per interval on both axes, plot the four points and explain why they should remain separate points.
4 marks
Label the horizontal axis Stage $n$ and the vertical axis Tiles $T$ , with $1$ per interval. (1 mark)
Correctly plot $(1, 3)$ and $(2, 5)$ . (1 mark)
Correctly plot $(3, 7)$ and $(4, 9)$ . (1 mark)
Use separate points: fractional stage numbers do not represent stages of this tile pattern. (1 mark)
Plot $(1, 3)$ , $(2, 5)$ , $(3, 7)$ and $(4, 9)$ on labelled axes with the stated scales. Keep the points separate because only positive whole-number stages exist.
Which stage of $T = 2 n + 1$ has $25$ tiles? Show your calculation and check your answer by substitution.
3 marks
$2 n + 1 = 25$ , so subtract $1$ from both sides to get $2 n = 24$ . (1 mark)
Divide both sides by $2$ : $n = 12$ . (1 mark)
Check: $2 (12) + 1 = 25$ . The stage number is a positive whole number. (1 mark)
Stage $12$ .
The horizontal axis is stage number and the vertical axis is number of tiles. Which point represents stage $7$ of $T = 2 n + 1$ : $(7, 15)$ or $(15, 7)$ ? Explain the coordinate order.
2 marks
$T = 2 (7) + 1 = 15$ , so choose $(7, 15)$ . (1 mark)
The first coordinate is horizontal, $n = 7$ ; the second is vertical, $T = 15$ . (1 mark)
$(7, 15)$ ; stage number comes first and total tiles second.
A different growing pattern has the rule $B = 3 n + 1$ . Make a table for stages $n = 1, 2, 3, 4$ , list the four coordinates $(n, B)$ and state how the total changes when the stage increases by $1$ .
4 marks
For stages $1$ and $2$ , $B = 3 (1) + 1 = 4$ and $B = 3 (2) + 1 = 7$ . (1 mark)
For stages $3$ and $4$ , $B = 3 (3) + 1 = 10$ and $B = 3 (4) + 1 = 13$ . (1 mark)
Pair each stage with its total: $(1, 4)$ , $(2, 7)$ , $(3, 10)$ and $(4, 13)$ . (1 mark)
The successive differences are $7 - 4 = 3$ , $10 - 7 = 3$ and $13 - 10 = 3$ , so the total increases by $3$ for each extra stage. (1 mark)
The totals are $4, 7, 10, 13$ . The points are $(1, 4)$ , $(2, 7)$ , $(3, 10)$ and $(4, 13)$ . The total increases by $3$ per stage.

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Curriculum reference & lesson scope

Australian Curriculum, presented by QCAA · 9.0

AC9M7A05: Growing patterns, tables and Cartesian graphs

Generate values from a visual pattern or function rule, describe the relationship and represent it with Cartesian coordinates.

Official curriculum source ↗

Source checked 2026-11-02. Schools may sequence content differently.

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