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 next patterns for each sequence: 1. Step 1: Count the number of circles in each pattern. The sequence is 1, 3, 6, 10. Step 2: Identify the pattern rule. This is a sequence of triangular numbers, where the n-th term is T_n = (n(n+1))/(2). The given patterns correspond to T_1, T_2, T_3, T_4. Step 3: Calculate the next term. The next term is T_5 = (5(5+1))/(2) = (5 × 6)/(2) = 15. Step 4: Describe the next pattern. The next pattern will have 15 circles arranged in a solid triangle with 5 circles at the base. Draw a triangle of 15 circles, with 5 circles in the bottom row, 4 in the next, 3, 2, and 1 at the top. 2. Step 1: Count the number of circles in each pattern. The sequence is 3, 6, 10, 15. Step 2: Identify the pattern rule. This is a sequence of triangular numbers, starting from T_2. The given patterns correspond to T_2, T_3, T_4, T_5. Step 3: Calculate the next term. The next term is T_6 = (6(6+1))/(2) = (6 × 7)/(2) = 21. Step 4: Describe the next pattern. The next pattern will have 21 circles arranged in a solid triangle with 6 circles at the base. Draw a triangle of 21 circles, with 6 circles in the bottom row, 5 in the next, 4, 3, 2, and 1 at the top. 3. Step 1: Count the number of circles in each pattern. The sequence is 1, 3, 6, 10. Step 2: Identify the pattern rule. This is a sequence of triangular numbers, where the n-th term is T_n = (n(n+1))/(2). The given patterns correspond to T_1, T_2, T_3, T_4. Step 3: Calculate the next term. The next term is T_5 = (5(5+1))/(2) = (5 × 6)/(2) = 15. Step 4: Describe the next pattern. The next pattern will have 15 circles arranged in a staircase shape, with 5 circles in the bottom row, 4 in the row above it, 3, 2, and 1 at the top. Draw a staircase pattern of 15 circles, with 5 circles in the bottom row, 4 in the row above, 3, 2, and 1 at the top. 4. Step 1: Count the number of dots in each pattern. The sequence is 1, 3, 6, 10. Step 2: Identify the pattern rule. This is a sequence of triangular numbers, where the n-th term is T_n = (n(n+1))/(2). The given patterns correspond to T_1, T_2, T_3, T_4, representing a triangle with n dots on each side. Step 3: Calculate the next term. The next term is T_5 = (5(5+1))/(2) = (5 × 6)/(2) = 15. Step 4: Describe the next pattern. The next pattern will have 15 dots. It will form a large triangle with 5 dots on each side, and all dots connected by lines to form smaller triangles within. Draw a large triangle with 5 dots on each side. Connect all dots with lines to form a grid of smaller triangles, resulting in 15 dots in total. That's 2 down. 3 left today — send the next one.