Question

In mathematics, relations can be expressed in a number of 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 given pattern and answer the questions that follow.

1. Write the AP for the number of triangles used in the figures. Also, write the (n)th term of this AP.
2. Write the AP for the number of matchsticks used in the figures. Also, write the (n)th term of this AP.
3. Find the total number of triangles used in the first 20 figures.
4. Which figure has 61 matchsticks?

Answer 1

Examining the figures:

Figure 1: 4 triangles, 12 matchsticks

Figure 2: 6 triangles, 19 matchsticks

Figure 3: 8 triangles, 26 matchsticks

i. AP for the number of triangles and its (n)-th term:

Sequence: \[4, 6, 8, \ldots\]

First term \[a = 4\], common difference \[d = 6 - 4 = 2\]

(n)-th term: \[T_n = a + (n - 1)d\]

\[T_n = 4 + (n - 1)2\]

\[= 4 + 2n - 2\]

\[= 2n + 2\]

AP is \[4, 6, 8, \ldots\] and (n)-th term is \[2n + 2\].

ii. AP for the number of matchsticks and its (n)-th term:

Sequence: \[12, 19, 26, \ldots\]

First term \[a = 12\], common difference \[d = 19 - 12 = 7\]

(n)-th term: \[T_n = a + (n - 1)d\]

\[T_n = 12 + (n - 1)7\]

\[= 12 + 7n - 7\]

\[= 7n + 5\]

AP is \[12, 19, 26, \ldots\] and (n)-th term is \[7n + 5\].

iii. Total number of triangles used in the first 20 figures:

Formula: \[S_n = \frac{n}{2}[2a + (n - 1)d]\] 

For triangles: \[a = 4, d = 2, n = 20\]

\[S_{20} = \frac{20}{2}[2(4) + (20 - 1)2]\]

\[S_{20} = 10 \times [8 + 19 \times 2]\]

\[S_{20} = 10 \times [8 + 38]\]

\[= 10 \times 46\]

\[= 460\]

Total triangles used is 460.

iv. Figure having 61 matchsticks:

Set the (n)-th term of matchsticks to 61:

\[7n + 5 = 61\]

\[7n = 61 - 5\]

\[7n = 56\]

\[n = \frac{56}{7} = 8\]

Figure 8 has 61 matchsticks.