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.
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Here are the solutions to the pattern problems. 1.1.1 Sticks Pattern a) Step 1: Observe the pattern of sticks. The number of sticks for the first three patterns are 4, 7, 10. This is an arithmetic sequence with a common difference d = 7 - 4 = 3. Step 2: Calculate the number of sticks for the next pattern. The next pattern (4th pattern) will have 10 + 3 = 13 sticks. The number of sticks needed is 13 sticks. b) The number of sticks increases by 3 for each additional square added to the pattern. c) Step 1: Complete the table using the common difference of 3. |c|c| Pattern number & Number of sticks \\ 1 & 4 \\ 2 & 7 \\ 3 & 10 \\ 4 & 13 \\ 5 & 16 \\ 6 & 19 \\ 7 & 22 \\ 8 & 25 \\ d) Step 1: Identify the first term (a) and common difference (d). a = 4, d = 3. Step 2: Use the formula for the n-th term of an arithmetic sequence, T_n = a + (n-1)d. For the 18th pattern (n=18): T_18 = 4 + (18-1) × 3 Step 3: Calculate the value. T_18 = 4 + 17 × 3 = 4 + 51 = 55 The number of sticks needed for the 18th pattern is 55 sticks. e) Step 1: Use the formula for the n-th term of an arithmetic sequence, T_n = a + (n-1)d. Substitute a=4 and d=3. T_n = 4 + (n-1) × 3 Step 2: Simplify the expression. T_n = 4 + 3n - 3 = 3n + 1 The algebraic rule is T_n = 3n + 1. 1.1.2 Geometric shape pattern a) Step 1: Analyze the pattern of dots. Pattern 1 has 4 × 4 = 16 dots. Pattern 2 has 3 × 3 = 9 dots. Pattern 3 has 2 × 2 = 4 dots. Pattern 4 has 1 × 1 = 1 dot. Step 2: Determine the rule for the number of dots (T_n) in terms of the pattern number (n). The side length of the square is 5-n. So, T_n = (5-n)^2. Step 3: Calculate the number of dots for the 8th term (n=8). T_8 = (5-8)^2 Step 4: Simplify the expression. T_8 = (-3)^2 = 9 The number of dots in the 8th term is 9 dots. b) Step 1: Use the rule T_n = (5-n)^2 for the 10th term (n=10). T_10 = (5-10)^2 Step 2: Simplify the expression. T_10 = (-5)^2 = 25 The 10th term of the pattern is 25 dots. c) Step 1: Based on the analysis in part (a), the algebraic rule is the square of (5-n). The algebraic rule is T_n = (5-n)^2. That's 2 down. 3 left today — send the next one.