The patterns show a sequence of shapes. Each shape has a number of small right-angled triangles. A dot is placed at each point where there is a vertex of one or more right-angled triangles. The total number of dots and the number of small right-angled triangles of each figure is shown in a table.

1. Find the total number of dots required to form Pattern 6.
2. Find the total number of dots required to form Pattern 47.
3. Find the number of small right-angled triangles in Pattern 40.

(a)
Pattern for the total number of dots
Pattern 1: (1 + 1)² = 4
Pattern 2: (2 + 1)² = 9
Pattern 3: (3 + 1)² = 16

Formula
Number of dots = (Pattern number + 1)²

Number of dots for Pattern 6
= (6 + 1)²
= 49

(b)
Number of dots in Pattern 47
= (47 + 1)²
= 48²
= 2304

(c)
Pattern for the number of small right-angled triangles
Pattern 1: 1² x 2 = 2
Pattern 2: 2² x 2 = 8
Pattern 3: 3² x 2 = 18

Formula
Number of small right-angled triangles = Pattern number² x 2

Number of small right-angled triangles in Pattern 40
= 40² x 2
= 3200

Answer(s):(a) 49; (b) 2304; (c) 3200