Look at the first three stages of a growing pattern of hexagons made using matchsticks.

Question

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.

(A) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

(B) Complete the following table.

Stage Number	Number of matchsticks
$1$
$2$
$3$
$4$
$5$
$...$
$n$

(C) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

(D) How many matchsticks will be required for the $15^{\text{th}}$ stage of the pattern?

(E) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Educart · Accepted Answer

Stage 1: one hexagon 6 matchsticks.

Stage 2: two hexagons sharing a side $\rightarrow$ $6 + 5 = 11$ matchsticks.

Stage 3: three hexagons $\rightarrow$ $11 + 5 = 16$ matchsticks.

Each new stage adds 5 matchsticks.

Stage 4: $4 \times 5 + 1 = 21$ matchsticks.

Stage 5: $21 + 5 = 26$ matchsticks

(A)

(B) Table of values:

Stage number	Number of Matchsticks
1	6
2	11
3	16
4	21
5	26
...	...
$n$	$5n + 1$

(C) General rule: Number of matchsticks = $5n + 1$

(D) $15^{\text{th}}$ stage: $5(15) + 1 = 76$ matchsticks.

(E) $5n + 1 = 200$

$\Rightarrow 5n = 199$

$\Rightarrow n = 39.8$ , which is not a whole number.

So, 200 matchsticks cannot exactly form a stage of this pattern.

To Download PDF