In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in diffe

Question

In Mathematics, relations can be expressed in various ways. The matchstick patterns are based on linear relations. Different strategies can be used to calculate the number of matchsticks used in different figures.

One such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression:

[IMAGE_1]
[IMAGE_2]

(a) Write the AP for the number of triangles used in the figures. Also, write the $n^{th}$ term of this AP.

(b) Which figure has $61$ matchsticks?

Accepted Answer

Step 1: (a) Number of triangles in figures are $4, 6, 8, ...$. This is an A.P. with $a = 4$, $d = 2$.
Step 2: $a_n = 4 + (n - 1) 	imes 2 = 2n + 2$
Step 3: (b) Number of matchsticks in figures are $12, 19, 26, ...$. This is an A.P. with $a = 12$, $d = 7$.
Step 4: $61 = 12 + (n - 1) 	imes 7 \Rightarrow 49 = (n - 1) 	imes 7 \Rightarrow n - 1 = 7 \Rightarrow n = 8$
Step 5: Figure $8$ has $61$ matchsticks.

Question

Solution