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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1.1 What is a linear number pattern? A linear number pattern, also known as an arithmetic sequence, is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. 1.2 Explain what a general term (or n^th term) of a sequence is. The general term (or n^th term) of a sequence is a formula, usually denoted as T_n or a_n, that allows you to calculate any term in the sequence by substituting its position number n. 1.3 What is the difference between a term number and a term value? The term number refers to the position of a term in a sequence (e.g., 1st, 2nd, 3rd term), while the term value is the actual numerical value of the term at that specific position. --- Question 2: Step 1: Identify the number of dots in each figure and determine the common difference. Figure 1 (T_1) has 6 dots. Figure 2 (T_2) has 9 dots. Figure 3 (T_3) has 12 dots. The common difference d is: d = T_2 - T_1 = 9 - 6 = 3 d = T_3 - T_2 = 12 - 9 = 3 The common difference is 3. 2.1 Determine the general term of the number pattern above. The formula for the general term of a linear pattern is T_n = a + (n-1)d, where a is the first term and d is the common difference. Substitute a=6 and d=3: T_n = 6 + (n-1)3 T_n = 6 + 3n - 3 T_n = 3n + 3 The general term is T_n = 3n + 3. 2.2 How many dots will be needed to build the 10th figure? To find the number of dots for the 10th figure, substitute n=10 into the general term T_n = 3n + 3: T_10 = 3(10) + 3 T_10 = 30 + 3 T_10 = 33 The 10th figure will need 33 dots. 2.3 116 dots are needed for the k^th figure. Calculate the value of k. Set the general term T_k equal to 116 and solve for k: 3k + 3 = 116 Subtract 3 from both sides: 3k = 116 - 3 3k = 113 Divide by 3: k = (113)/(3) k ≈ 37.67 The value of k is (113)/(3). Since k represents a figure number, it is typically expected to be a positive integer. As (113)/(3) is not an integer, 116 dots cannot form an exact figure in this pattern. Send me the next one 📸