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. The figures show triangles made of matchsticks: Figure 1 (1 triangle), Figure 2 (2 triangles), Figure 3 (3 triangles), and so on.
Observe the pattern and answer the following questions using Arithmetic Progression:
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding stimulus
Model Answer
(a)
Number of triangles in each figure: 1, 2, 3, 4, …
This is an AP with first term a = 1 and common difference d = 1.
The *n*th term: $a_n = a + (n-1)d = 1 + (n-1)(1) = n$
So, $a_n = n$
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(b)
For triangles in a row, matchsticks used = $2n + 1$ (each new triangle adds 2 matchsticks to the previous figure, starting with 3).
AP for matchsticks: 3, 5, 7, … with a = 3, d = 2.
Given $a_n = 61$:
$$a + (n-1)d = 61$$
$$3 + (n-1)(2) = 61$$
$$(n-1)(2) = 58$$
$$n - 1 = 29 \Rightarrow n = 30$$
∴ Figure 30 has 61 matchsticks.
Source: Arithmetic Progressions, Linear Relations through Matchstick Patterns
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Explanation
- Part (a) asks for the AP of number of triangles (1, 2, 3…), not matchsticks — keep them separate.
- Part (b) uses the matchstick count AP: Figure 1 uses 3, Figure 2 uses 5, Figure 3 uses 7 (each new triangle shares one side, so adds 2 sticks). Formula $a_n = 2n+1$ must be derived from this pattern.
- Always show the substitution step clearly — it earns method marks.