Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.
- (i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?
- (ii) Complete the following table.
- (iii) Find a rule to determine the number of matchsticks required for the nth stage.
- (iv) How many matchsticks will be required for the 15th stage of the pattern?
- (v) Can 200 matchsticks form a stage in this pattern? Justify your answer.
Step-by-step solution
Idea: The first hexagon needs 6 matchsticks. Each new hexagon shares one side with the previous one, so it needs only 5 new matchsticks. The counts go up by 5 each stage: a linear pattern.
(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?
- Stage 4 and Stage 5 are drawn in the picture: add one more hexagon each time, sharing a side with the last one.½ mark
- Stage 1: 6; each new hexagon adds 5. Stage 4 = 6 + 3 × 5 = 21; Stage 5 = 21 + 5 = 26 matchsticks.½ mark
(ii) Complete the following table.
- 1 mark
Stage Number 1 2 3 4 5 … n Number of matchsticks 6 11 16 21 26 … 5n + 1
(iii) Find a rule to determine the number of matchsticks required for the nth stage.
- The first hexagon uses 6; each of the other n − 1 hexagons adds 5. Matchsticks = 6 + 5(n − 1) = 5n + 1.1 mark
(iv) How many matchsticks will be required for the 15th stage of the pattern?
- 5 × 15 + 1 = 75 + 1 = 76 matchsticks.1 mark
(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.
- 5n + 1 = 200 ⇒ 5n = 199 ⇒ n = 39.8, which is not a whole number. (Stage 39 uses 196 and Stage 40 uses 201.) So no, 200 matchsticks cannot form a stage exactly.1 mark
Check: 5n + 1 at n = 1, 2, 3 gives 6, 11, 16, which match the three stages in the figure ✓. Also, 5n + 1 always leaves remainder 1 when divided by 5, and 200 leaves remainder 0.
Answer to write in the exam
(i)
Stages 4 and 5 drawn
Each new hexagon needs 5 more matchsticks
∴ Stage 4: 6 + 3 × 5 = 21 matchsticks; Stage 5: 21 + 5 = 26 matchsticks
(ii)
Stage: 1, 2, 3, 4, 5, …, n
∴ Matchsticks: 6, 11, 16, 21, 26, …, 5n + 1
(iii)
Matchsticks at Stage n = 6 + 5(n − 1)
∴ Rule: 5n + 1
(iv)
5(15) + 1 = 75 + 1
∴ 76 matchsticks
(v)
5n + 1 = 200 ⇒ 5n = 199 ⇒ n = 39.8
n must be a natural number
∴ No, 200 matchsticks cannot form a stage
Common mistakes that cost marks
- Counting 6 matchsticks for every hexagon (6n). Shared sides are counted only once, so each new hexagon adds 5.
- Writing the rule as 5n (forgetting the extra stick of the first hexagon) or 5n + 6.
- Answering (v) with “yes, about Stage 40”. A stage number must be a whole number.
How this can come in the exam
Pentagons are made in a row with matchsticks, each new pentagon sharing a side with the previous one: 5, 9, 13, … matchsticks. The rule for the nth stage is
- 5n
- 4n + 1
- 4n + 5
- n + 4
Show answer
(B) 4n + 1
First pentagon 5, then 4 more each time: 5 + 4(n − 1) = 4n + 1.
A row of triangles is made with matchsticks, each new triangle sharing a side with the previous one: 3, 5, 7, … matchsticks.
(i) Find the number for stage 6. (ii) Write the rule for stage n. (iii) How many are needed for stage 25? (iv) Can 100 matchsticks form a stage?
Show answer
(i) 13 (1 mark). (ii) 2n + 1 (1 mark). (iii) 51 (1 mark). (iv) 2n + 1 = 100 ⇒ n = 49.5, not whole, so no (1 mark).Try one yourself
In the hexagon pattern, which stage uses 101 matchsticks?
Show answer
5n + 1 = 101 ⇒ n = 20: Stage 20.
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