Class 9th Maths Ganita Manjari - Part I Chapter 2: Introduction to Linear Polynomials Exercise 7: End-Of-Chapter Exercises Question 12 (i) Solution

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.

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

Three stages of a growing hexagon pattern made using matchsticks, with each new hexagon sharing a side with the previous one. — First three stages of a growing hexagon pattern made using matchsticks

Accepted Answer

Count the matchsticks in the given stages.
 	ext{Stage 1 = 6 matchsticks}|||	ext{Stage 2 = 6 + 5 = 11 matchsticks}|||	ext{Stage 3 = 6 + 5 + 5 = 16 matchsticks} Find how the number of matchsticks increases from one stage to the next.
 	ext{Stage 2 - Stage 1 = 11 - 6 = 5}|||	ext{Stage 3 - Stage 2 = 16 - 11 = 5} Determine the linear expression{math_tip:def_linear_expression} for the number of matchsticks in the $n$th stage.
 	ext{Matchsticks in Stage 1 = 6}|||	ext{Matchsticks added at each new stage = 5 $\quad$ ${color:#625B5F}	extit{[For example, Stage 2 = 6 + 5 = 11 matchsticks]}{/color}$}|||	ext{Hence, the number of matchsticks $c$ in the $n$th stage is $c = 6 + 5(n - 1)$} Find the number of matchsticks in the 4th and 5th stages.
 	ext{Matchsticks in the 4th stage:}|||	ext{marker{\Rightarrow} $6 + 5 × ({color:#2E2EFF}4{/color} - 1)$}|||	ext{marker{\Rightarrow} $6 + 5 × 3$}|||	ext{marker{\Rightarrow} $6 + 15 = 21$}|||	ext{Matchsticks in the 5th stage:}|||	ext{marker{\Rightarrow} $6 + 5 × ({color:#2E2EFF}5{/color} - 1)$}|||	ext{marker{\Rightarrow} $6 + 5 × 4$}|||	ext{marker{\Rightarrow} $6 + 20 = 26$} Draw the pattern for the next two stages.

Question: 12 (i)

Answer