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?

SOLUTION

Understand the pattern

A single hexagon uses \(6\) matchsticks. Every new hexagon shares \(1\) side with the previous hexagon, so a new hexagon adds only \(6-1 = 5\) matchsticks.
There is an increase of \(5\) matchsticks each stage.

Count the number of matchsticks required for the next two stages

Stage 1: \(1\) hexagon requires \(6\) matchsticks.
Stage 2: A second hexagon is added sharing one side with the first. So, \(6+5 = 11\) matchsticks.
Stage 3: For third hexagon, \(11+5 = 16\) matchsticks.
Stage 4: For fourth hexagon, \(16+5 = 21\) matchsticks.
Stage 5: For fifth hexagon, \(21+5 = 26\) matchsticks.

Draw the next two stages of the pattern

Shown in figure.

Concept Note

The number of matchsticks increases by a fixed amount each stage, so the relationship is linear.

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?

SOLUTION

Concept Note

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?