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 tackle this arithmetic series problem. Question 2 2.1.1 Determine the value of x. Step 1: Identify the given information for the arithmetic series. The common difference is d = 4. The fourth term is T_4 = 3x-1. The seventh term is T_7 = 2x+8. The formula for the n-th term of an arithmetic series is T_n = a + (n-1)d. Step 2: Express T_7 in terms of T_4 and the common difference. The difference between the seventh term and the fourth term is 3d. T_7 = T_4 + (7-4)d T_7 = T_4 + 3d Step 3: Substitute the given expressions and the common difference into the equation. 2x+8 = (3x-1) + 3(4) 2x+8 = 3x-1 + 12 2x+8 = 3x+11 Step 4: Solve for x. Subtract 2x from both sides: 8 = x+11 Subtract 11 from both sides: 8-11 = x x = -3 The value of x is -3. 2.1.2 Calculate the: a) First term of the series Step 1: Use the value of x to find T_4. T_4 = 3x-1 T_4 = 3(-3)-1 T_4 = -9-1 T_4 = -10 Step 2: Use the formula T_n = a + (n-1)d to find the first term a. We know T_4 = -10, n=4, and d=4. T_4 = a + (4-1)d -10 = a + 3(4) -10 = a + 12 Step 3: Solve for a. a = -10 - 12 a = -22 The first term of the series is -22. b) Sum of the first 42 terms of the series Step 1: Identify the values needed for the sum formula. First term a = -22 (from 2.1.2a). Common difference d = 4. Number of terms n = 42. The formula for the sum of the first n terms of an arithmetic series is S_n = (n)/(2)[2a + (n-1)d]. Step 2: Substitute the values into the sum formula. S_42 = (42)/(2)[2(-22) + (42-1)4] S_42 = 21[-44 + (41)4] S_42 = 21[-44 + 164] S_42 = 21[120] Step 3: Calculate the sum. S_42 = 2520 The sum of the first 42 terms of the series is 2520. That's 2 down. 3 left today — send the next one.