This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the extended patterns and their descriptions: Patterns in a sequence (Triangles) Step 1: Analyze the visual pattern of triangles. • Pattern 1: 1 purple triangle + 1 green triangle = 2 triangles. • Pattern 2: 2 purple triangles + 2 green triangles = 4 triangles. • Pattern 3: 3 purple triangles + 3 green triangles = 6 triangles. • Pattern 4: 4 purple triangles + 4 green triangles = 8 triangles. The number of triangles in pattern n is 2n. Step 2: Create a table to show the number of blocks up to the tenth pattern. | Pattern Number (n) | Number of Blocks (2n) | | :------------------- | :---------------------- | | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 | | 5 | 10 | | 6 | 12 | | 7 | 14 | | 8 | 16 | | 9 | 18 | | 10 | 20 | Step 3: State the tenth pattern. The tenth pattern will have 2 × 10 = 20 triangles. The answer is The tenth pattern has 20 triangles. e. Pattern of dots Step 1: Analyze the pattern of dots. • Pattern 1: 3 dots • Pattern 2: 6 dots • Pattern 3: 10 dots • Pattern 4: 15 dots This is a sequence of triangular numbers. The number of dots in pattern n is given by the formula ((n+1)(n+2))/(2). Step 2: Extend the pattern to the 5th pattern. For the 5th pattern (n=5): Number of dots = ((5+1)(5+2))/(2) = (6 × 7)/(2) = (42)/(2) = 21. The 5th pattern will have 21 dots arranged in a triangle with 6 dots on each side. f. Pattern of houses Step 1: Analyze the pattern of houses. • Pattern 1: 1 triangle, 3 squares. Total = 4 blocks. • Pattern 2: 2 triangles, 6 squares. Total = 8 blocks. • Pattern 3: 3 triangles, 9 squares. Total = 12 blocks. • Pattern 4: 4 triangles, 12 squares. Total = 16 blocks. The number of blocks in pattern n is 4n. Step 2: Extend the pattern to the 5th pattern. For the 5th pattern (n=5): Number of blocks = 4 × 5 = 20. The 5th pattern will consist of 5 houses, each with 1 triangle and 3 squares, for a total of 5 triangles and 15 squares. 2. Extend the patterns. a) Step 1: Analyze the pattern of colored squares. • Pattern 1: A 2 × 2 square, total 4 small squares. • Pattern 2: A 4 × 4 square, total 16 small squares. The side length of the square is 2^n. The total number of small squares is (2^n)^2 = 2^2n. Step 2: Extend the pattern to the 3rd pattern. For the 3rd pattern (n=3): Side length = 2^3 = 8. Total number of small squares = 8 × 8 = 64. The 3rd pattern will be an *$8