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's how to solve this problem: a) Find the common difference d. Step 1: Write the given information using the formula for the n-th term of an arithmetic sequence, T_n = a + (n-1)d. The 4th term is 19: T_4 = a + (4-1)d 19 = a + 3d (Equation 1) The 9th term is 44: T_9 = a + (9-1)d 44 = a + 8d (Equation 2) Step 2: Subtract Equation 1 from Equation 2 to eliminate a and solve for d. (a + 8d) - (a + 3d) = 44 - 19 5d = 25 Step 3: Solve for d. d = (25)/(5) d = 5 Answer: d = 5 b) Find the first term a. Step 1: Substitute the value of d=5 into Equation 1. 19 = a + 3(5) 19 = a + 15 Step 2: Solve for a. a = 19 - 15 a = 4 Answer: a = 4 c) Write the first six terms of the sequence. Step 1: Use the first term a=4 and the common difference d=5 to find the terms. T_1 = a = 4 T_2 = a + d = 4 + 5 = 9 T_3 = a + 2d = 4 + 2(5) = 4 + 10 = 14 T_4 = a + 3d = 4 + 3(5) = 4 + 15 = 19 T_5 = a + 4d = 4 + 4(5) = 4 + 20 = 24 T_6 = a + 5d = 4 + 5(5) = 4 + 25 = 29 Answer: 4, 9, 14, 19, 24, 29 3 done, 2 left today. You're making progress.