Sec 1 Maths · Number sequences and patterns

Patterns questions with answers

Practice questions on patterns, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

The diagram shows Figures 3, 4 and 5 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 6.

Step 1 of 3

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

\text{Figure 3:}

10

\text{Figure 4:}

13

\text{Figure 5:}

16

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Questions

Question 1Continue the pattern
The diagram shows Figures 1, 2 and 3 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 5.
Show the worked solution
1. Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 1:}$ $4$ $\text{Figure 2:}$ $7$ $\text{Figure 3:}$ $10$
2. Step 2: How many more matchsticks does each new figure need? $\mathstrut 7 - 4 =$ $3$ Figure $1, 2, 3$ : matchsticks $4, 7, 10$ , going up by $3$ each time.
3. Step 3: Add the difference once for each figure after the last one drawn $\mathstrut 10 + 2 \times 3 =$ $16$
Question 2Continue the pattern
The diagram shows Figures 3, 4 and 5 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 7.
Show the worked solution
1. Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 3:}$ $10$ $\text{Figure 4:}$ $13$ $\text{Figure 5:}$ $16$
2. Step 2: How many more matchsticks does each new figure need? $\mathstrut 13 - 10 =$ $3$ Figure $3, 4, 5$ : matchsticks $10, 13, 16$ , going up by $3$ each time.
3. Step 3: Add the difference once for each figure after the last one drawn $\mathstrut 16 + 2 \times 3 =$ $22$
Question 3Matchsticks in Figure n
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 $n$ .
Show the worked solution
1. Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 2:}$ $12$ $\text{Figure 3:}$ $17$ $\text{Figure 4:}$ $22$
2. Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 17 - 12 =$ $5$ Figure $2, 3, 4$ : matchsticks $12, 17, 22$ , going up by $5$ each time.
3. Step 3: Compare with (difference)n: from the first figure's count, take away (its figure number) × (difference) $\mathstrut 12 - 2 \times 5 =$ $2$
4. Step 4: So Figure n needs $5n + 2$ matchsticks
Question 4Matchsticks in Figure n
The diagram shows Figures 1, 2 and 3 of a pattern made with matchsticks. Find an expression, in terms of $n$ , for the number of matchsticks in Figure $n$ .
Show the worked solution
1. Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 1:}$ $6$ $\text{Figure 2:}$ $11$ $\text{Figure 3:}$ $16$
2. Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 11 - 6 =$ $5$ Figure $1, 2, 3$ : matchsticks $6, 11, 16$ , going up by $5$ each time.
3. Step 3: Compare with (difference)n: from the first figure's count, take away (its figure number) × (difference) $\mathstrut 6 - 5 =$ $1$
4. Step 4: So Figure n needs $5n + 1$ matchsticks
Question 5A far-away figure
The diagram shows Figures 1, 2 and 3 of a pattern made with matchsticks. Figure $n$ uses exactly $46$ matchsticks. Find the value of $n$ .
Show the worked solution
1. Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 1:}$ $4$ $\text{Figure 2:}$ $7$ $\text{Figure 3:}$ $10$
2. Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 7 - 4 =$ $3$ Figure $1, 2, 3$ : matchsticks $4, 7, 10$ , going up by $3$ each time.
3. Step 3: The number of matchsticks in Figure n (it goes up by the same amount each time) $3n + 1$ , since Figures $1, 2, 3$ have $4, 7, 10$ matchsticks
4. Step 4: Set it equal to 46 and solve $3n + 1 = 46$ $3n = 45$ $n = 15$ Figure $15$
Question 6A far-away figure
The diagram shows Figures 2, 3 and 4 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 100.
Show the worked solution
1. Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 2:}$ $11$ $\text{Figure 3:}$ $16$ $\text{Figure 4:}$ $21$
2. Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 16 - 11 =$ $5$ Figure $2, 3, 4$ : matchsticks $11, 16, 21$ , going up by $5$ each time.
3. Step 3: The number of matchsticks in Figure n (it goes up by the same amount each time) $5n + 1$ , since Figures $2, 3, 4$ have $11, 16, 21$ matchsticks
4. Step 4: Put n = 100 $\mathstrut 5 \times 100 + 1 =$ $501$

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Worked example

The diagram shows Figures 3, 4 and 5 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 6.

Step 1 of 3

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

\text{Figure 3:}

10

\text{Figure 4:}

13

\text{Figure 5:}

16

Questions

Question 1Continue the pattern

The diagram shows Figures 1, 2 and 3 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 5.

Show the worked solution

Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 1:}$ $4$ $\text{Figure 2:}$ $7$ $\text{Figure 3:}$ $10$
Step 2: How many more matchsticks does each new figure need? $\mathstrut 7 - 4 =$ $3$ Figure $1, 2, 3$ : matchsticks $4, 7, 10$ , going up by $3$ each time.
Step 3: Add the difference once for each figure after the last one drawn $\mathstrut 10 + 2 \times 3 =$ $16$

Question 2Continue the pattern

The diagram shows Figures 3, 4 and 5 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 7.

Show the worked solution

Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 3:}$ $10$ $\text{Figure 4:}$ $13$ $\text{Figure 5:}$ $16$
Step 2: How many more matchsticks does each new figure need? $\mathstrut 13 - 10 =$ $3$ Figure $3, 4, 5$ : matchsticks $10, 13, 16$ , going up by $3$ each time.
Step 3: Add the difference once for each figure after the last one drawn $\mathstrut 16 + 2 \times 3 =$ $22$

Question 3Matchsticks in Figure n

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 $n$ .

Show the worked solution

Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 2:}$ $12$ $\text{Figure 3:}$ $17$ $\text{Figure 4:}$ $22$
Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 17 - 12 =$ $5$ Figure $2, 3, 4$ : matchsticks $12, 17, 22$ , going up by $5$ each time.
Step 3: Compare with (difference)n: from the first figure's count, take away (its figure number) × (difference) $\mathstrut 12 - 2 \times 5 =$ $2$
Step 4: So Figure n needs $5n + 2$ matchsticks

Question 4Matchsticks in Figure n

The diagram shows Figures 1, 2 and 3 of a pattern made with matchsticks. Find an expression, in terms of $n$ , for the number of matchsticks in Figure $n$ .

Show the worked solution

Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 1:}$ $6$ $\text{Figure 2:}$ $11$ $\text{Figure 3:}$ $16$
Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 11 - 6 =$ $5$ Figure $1, 2, 3$ : matchsticks $6, 11, 16$ , going up by $5$ each time.
Step 3: Compare with (difference)n: from the first figure's count, take away (its figure number) × (difference) $\mathstrut 6 - 5 =$ $1$
Step 4: So Figure n needs $5n + 1$ matchsticks

Question 5A far-away figure

The diagram shows Figures 1, 2 and 3 of a pattern made with matchsticks. Figure $n$ uses exactly $46$ matchsticks. Find the value of $n$ .

Show the worked solution

Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 1:}$ $4$ $\text{Figure 2:}$ $7$ $\text{Figure 3:}$ $10$
Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 7 - 4 =$ $3$ Figure $1, 2, 3$ : matchsticks $4, 7, 10$ , going up by $3$ each time.
Step 3: The number of matchsticks in Figure n (it goes up by the same amount each time) $3n + 1$ , since Figures $1, 2, 3$ have $4, 7, 10$ matchsticks
Step 4: Set it equal to 46 and solve $3n + 1 = 46$ $3n = 45$ $n = 15$ Figure $15$

Question 6A far-away figure

The diagram shows Figures 2, 3 and 4 of a pattern made with matchsticks. Find the number of matchsticks needed for Figure 100.

Show the worked solution

Step 1: Count the matchsticks in each figure (a side shared by two shapes is one matchstick) $\text{Figure 2:}$ $11$ $\text{Figure 3:}$ $16$ $\text{Figure 4:}$ $21$
Step 2: How many more matchsticks does each new figure need? It is the number in front of n $\mathstrut 16 - 11 =$ $5$ Figure $2, 3, 4$ : matchsticks $11, 16, 21$ , going up by $5$ each time.
Step 3: The number of matchsticks in Figure n (it goes up by the same amount each time) $5n + 1$ , since Figures $2, 3, 4$ have $11, 16, 21$ matchsticks
Step 4: Put n = 100 $\mathstrut 5 \times 100 + 1 =$ $501$