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.

Stage Number	1	2	3	4	5	…	n
Number of matchsticks

(iii) Find a rule to determine the number of matchsticks required for the n^th 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.

From the pattern:

Stage 1 has 1 hexagon, requiring 6 matchsticks.

Stage 2 has 2 hexagons; the second hexagon shares one side with the first, so it adds 5 matchsticks. Total = 6 + 5 = 11 matchsticks.

Stage 3 has 3 hexagons; each new hexagon adds 5 matchsticks. Total = 11 + 5 = 16 matchsticks.

So, each new stage adds 5 more matchsticks to the previous stage.

(i) Stage 4 has 4 hexagons. Total matchsticks = 16 + 5 = 21

Stage 5 has 5 hexagons. Total matchsticks = 21 + 5 = 26

Hence, the 4th stage requires 21 matchsticks and the 5th stage requires 26 matchsticks.

(ii) Stage 1: 6 matchsticks

Stage 2: 11 matchsticks

Stage 3: 16 matchsticks

Stage 4: 21 matchsticks

Stage 5: 26 matchsticks

The completed table is:

(iii) Each stage has 5 more matchsticks than the previous stage, starting with 6 in stage 1.

Stage 1: 6 = 5(1) + 1

Stage 2: 11 = 5(2) + 1

Stage 3: 16 = 5(3) + 1

Stage 4: 21 = 5(4) + 1

Stage 5: 26 = 5(5) + 1

In general, at the n^th stage, the number of matchsticks = 5n + 1.

Hence, the rule is: Number of matchsticks at n^th stage = 5n + 1.

(iv) Number of matchsticks at the 15th stage:

= 5(15) + 1

= 75 + 1

= 76

∴ The 15th stage requires 76 matchsticks.

(v) For 200 matchsticks to form a stage, we need:

5n + 1 = 200

⇒ 5n = 200 - 1

⇒ 5n = 199

⇒ n = $\dfrac{199}{5}$

⇒ n = 39.8

Hence, 200 matchsticks cannot form a stage in this pattern, since 39.8 is not a whole number.