Shapes in the form of small equilateral triangles can be made using matchsticks of equal length - Leaving Cert Mathematics - Question 9 - 2013

The completed table is as follows:

Pattern	1st	2nd	3rd	4th
Number of small triangles	1	9	25	49
Number of matchsticks	3	9	18	30

The expression for the number of small triangles in the nth pattern is:

$T_n = n^2$

This describes the quadratic growth of small triangles as the pattern progresses.

The expression for the number of matchsticks needed to transition from the (n-1)th pattern to the nth pattern is:

$M_n = 3n$

This indicates a linear relation between the pattern number and the matchsticks needed.

After analyzing the pattern, we establish:

• From the second differences of matchsticks: $$$$

ightarrow a = \frac{3}{2}$$

• Plugging values back, we find:
  $b = \frac{3}{2}$ Thus, we have determined:
• $a = \frac{3}{2}$
• $b = \frac{3}{2}$

To find how many small triangles are in the pattern with 4134 matchsticks, we solve:

$3n = 4134$

which leads to:

$n = \frac{4134}{3} \rightarrow n = 1378$

Now using the expression for small triangles:

$T_n = n^2 = 1378^2 = 1903684$

Hence, the number of small triangles in that pattern is:

1,904,684 triangles.