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
Okay, Kwanga, let's continue with Question 2.2. Question 2 2.2.1 Write down the second and third terms of the quadratic number pattern. Step 1: Identify the first term of the quadratic pattern and the formula for its first differences. The first term of the quadratic number pattern is Q_1 = 61. The first differences are given by D_k = 4k-26. Step 2: Calculate the first difference D_1. The first difference between the first and second terms is D_1. D_1 = 4(1) - 26 = 4 - 26 = -22 Step 3: Calculate the second term Q_2. The second term is the first term plus the first difference: Q_2 = Q_1 + D_1 Q_2 = 61 + (-22) Q_2 = 39 Step 4: Calculate the second difference D_2. The first difference between the second and third terms is D_2. D_2 = 4(2) - 26 = 8 - 26 = -18 Step 5: Calculate the third term Q_3. The third term is the second term plus the second difference: Q_3 = Q_2 + D_2 Q_3 = 39 + (-18) Q_3 = 21 The second term is 39 and the third term is 21. 2.2.2 If the n^th term of the quadratic number pattern is given by T_n = 2n^2 - 28n + 87, calculate the value of the smallest term. Step 1: Find the position (n-value) where the smallest term occurs. For a quadratic pattern T_n = an^2 + bn + c, the smallest (or largest) term occurs at n = (-b)/(2a). In the given formula T_n = 2n^2 - 28n + 87, we have a=2 and b=-28. n = (-(-28))/(2(2)) n = (28)/(4) n = 7 Since n=7 is an integer, the smallest term occurs at the 7th position. Step 2: Calculate the value of the smallest term by substituting n=7 into the formula. T_7 = 2(7)^2 - 28(7) + 87 T_7 = 2(49) - 196 + 87 T_7 = 98 - 196 + 87 T_7 = -98 + 87 T_7 = -11 The value of the smallest term is -11. 2.2.3 A constant k is added to T_n such that all the terms of the quadratic number pattern become positive. Determine the values of k. Step 1: Understand the effect of adding a constant k. If a constant k is added to each term T_n, the new terms are T_n' = T_n + k. For all terms to be positive, the smallest term of the new pattern must be greater than 0. Step 2: Use the smallest term found in 2.2.2. The smallest term of the original pattern is T_7 = -11. The smallest term of the new pattern will be T_7' = T_7 + k. Step 3: Set up the inequality for the smallest term to be positive. T_7 + k > 0 -11 + k > 0 Step 4: Solve for k. k > 11 The values of k are k > 11. 3 done, 2 left today. You're making progress.