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
3) Find the 10th term of the following sequences a) 8, 3, -2, -7, ... Step 1: Determine the type of sequence. Calculate the difference between consecutive terms: 3 - 8 = -5 -2 - 3 = -5 -7 - (-2) = -5 Since there is a common difference, d = -5, this is an arithmetic sequence. The first term is a_1 = 8. Step 2: Use the formula for the n-th term of an arithmetic sequence. The formula is a_n = a_1 + (n-1)d. We need to find the 10th term, so n=10. a_10 = 8 + (10-1)(-5) a_10 = 8 + (9)(-5) a_10 = 8 - 45 a_10 = -37 The 10th term is -37. b) 3, 1, (1)/(3), (1)/(9), ... Step 1: Determine the type of sequence. Calculate the ratio between consecutive terms: (1)/(3) (1/3)/(1) = (1)/(3) (1/9)/(1/3) = (1)/(9) × 3 = (3)/(9) = (1)/(3) Since there is a common ratio, r = (1)/(3), this is a geometric sequence. The first term is a_1 = 3. Step 2: Use the formula for the n-th term of a geometric sequence. The formula is a_n = a_1 r^n-1. We need to find the 10th term, so n=10. a_10 = 3 ((1)/(3))^10-1 a_10 = 3 ((1)/(3))^9 a_10 = 3 · (1^9)/(3^9) a_10 = 3 · (1)/(3^9) a_10 = (3)/(3^9) a_10 = (1)/(3^9-1) a_10 = (1)/(3^8) a_10 = (1)/(6561) The 10th term is (1)/(6561). 3 done, 2 left today. You're making progress.