**Q) ** 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:

(a) Write the AP for the number of triangles used in the figures. Also, write the nth term of this AP.

(b) Which figure has 61 matchsticks?

**Ans: a)**** AP and n ^{th} terms of the AP:**

No. of triangles used in figure 1 = 4

No. of triangles used in figure 2 = 6

No. of triangles used in figure 3 = 8 …….

Since the triangles are being added in a regular pattern, it should make an AP.

Let’s check the common differences between terms:

a_{3} – a_{2} = 8 – 6 = 2

a_{2} – a_{1} = 6 – 4 = 2

**Hence, it confirms this is an AP with terms as 4, 6, 8, …….**

Next, let’s find out n^{th} term:

Since n^{th} Term of an AP is given by:

a_{n} = a + (n – 1) d = 4 + (n – 1) x 2 = 2n + 2

**Hence, n ^{th} term of AP, a_{n} = 2n + 2**

**(b) Figure with 61 matchsticks:**

No. of sticks used in figure 1 = 12

No. of sticks used in figure 2 = 19

No. of sticks used in figure 3 = 26

Since the triangles are being added in a regular pattern, it should make an AP.

Let’s check the common differences between terms:

a_{3} – a_{2} = 26 – 19 = 7

a_{2} – a_{1} = 19 – 12 = 7

**Hence, it confirms this is an AP with terms as 12, 19, 26, …….**

Next, let’s find out n^{th} term:

Since n^{th} Term of an AP is given by:

a_{n} = a + (n – 1) d

and here, we are given that the n^{th} term is 61.

Therefore, a_{n} = 61 = 12 + (n – 1) x 7 = 5 + 7n

⇒ 7n = 61 – 5 = 56

⇒ n = = 8

**Therefore, 8 ^{th} figure will use 61 sticks.**