In Mathematics, relations can be expressed in various ways. The matchs

1.	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 figuresOne such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression:(a) Write the AP for the number of triangles used in the figures write the nth term of this AP(b) Which figure has 61 matchsticks?
Answer» (a) a₁ = number of sticks used in figure 1 = 12 = 12 + 6 x 0 a₂ = number of sticks used in figure 2 = 18 = 12 + 6 x 1 a₃ = number of sticks used in figure 3 = 24 = 12 + 6 x 2 ...................... a_n = number of sticks used in figure n = 12 + 6 x (n - 1) b₁ = number of triangle used in figure 1 = 4 b₂ = number of triangle used in figure 2 = 6 b₃ = number of triangle used in figure 3 = 8 d = b₂ - b₁ = 6 - 4 = 2 \(\therefore\) b_n = b₁ + (n - 1)d = 4 + (n - 1)2 = 2n + 2 A.P. for the number of triangles used in figure is 4, 6, 8... n^th term of A.P. is b_n = 2n + 2. (b) 12, 18, 24, .... is A.P. for the number of matchstick used in figures. d - a₂ - a₁ = 18 - 12 = 6 a_n = a + (n - 1)d = 12 + (n - 1)6 = 6n + 6 Let a_n = 61 then 6n + 6 = 61 ⇒ 6n = 61 - 6 = 56 ⇒ n = 56/6 = 9.33 Hence, 10^th figure has 61 match sticks.

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 figuresOne such pattern is shown below. Observe the pattern and answer the following questions using Arithmetic Progression:(a) Write the AP for the number of triangles used in the figures write the nth term of this AP(b) Which figure has 61 matchsticks?

Answer»

(a) a₁ = number of sticks used in figure 1 = 12 = 12 + 6 x 0

a₂ = number of sticks used in figure 2 = 18 = 12 + 6 x 1

a₃ = number of sticks used in figure 3 = 24 = 12 + 6 x 2

......................

a_n = number of sticks used in figure n = 12 + 6 x (n - 1)

b₁ = number of triangle used in figure 1 = 4

b₂ = number of triangle used in figure 2 = 6

b₃ = number of triangle used in figure 3 = 8

d = b₂ - b₁ = 6 - 4 = 2

\(\therefore\) b_n = b₁ + (n - 1)d

= 4 + (n - 1)2

= 2n + 2

A.P. for the number of triangles used in figure is 4, 6, 8... n^th term of A.P. is b_n = 2n + 2.

(b) 12, 18, 24, .... is A.P. for the number of matchstick used in figures.

d - a₂ - a₁ = 18 - 12 = 6

a_n = a + (n - 1)d

= 12 + (n - 1)6

= 6n + 6

Let a_n = 61

then 6n + 6 = 61

⇒ 6n = 61 - 6 = 56

⇒ n = 56/6 = 9.33

Hence, 10^th figure has 61 match sticks.