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 are the common differences and number of terms for each arithmetic progression: The general formula for the n-th term of an arithmetic progression (AP) is a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference, and n is the number of terms. The common difference d can be found using d = a_2 - a_1. a) 1, 5, ..., 69 Step 1: Find the common difference d. d = a_2 - a_1 = 5 - 1 = 4 Step 2: Find the number of terms n. a_n = a_1 + (n-1)d 69 = 1 + (n-1)4 68 = (n-1)4 (68)/(4) = n-1 17 = n-1 n = 17 + 1 n = 18 The common difference is 4 and the number of terms is 18. b) 80, 83, ..., 152 Step 1: Find the common difference d. d = a_2 - a_1 = 83 - 80 = 3 Step 2: Find the number of terms n. a_n = a_1 + (n-1)d 152 = 80 + (n-1)3 152 - 80 = (n-1)3 72 = (n-1)3 (72)/(3) = n-1 24 = n-1 n = 24 + 1 n = 25 The common difference is 3 and the number of terms is 25. c) 98, 91, ..., 0 Step 1: Find the common difference d. d = a_2 - a_1 = 91 - 98 = -7 Step 2: Find the number of terms n. a_n = a_1 + (n-1)d 0 = 98 + (n-1)(-7) -98 = (n-1)(-7) (-98)/(-7) = n-1 14 = n-1 n = 14 + 1 n = 15 The common difference is -7 and the number of terms is 15. d) 2.3, 3.4, ..., 21 Step 1: Find the common difference d. d = a_2 - a_1 = 3.4 - 2.3 = 1.1 Step 2: Find the number of terms n. a_n = a_1 + (n-1)d 21 = 2.3 + (n-1)1.1 21 - 2.3 = (n-1)1.1 18.7 = (n-1)1.1 (18.7)/(1.1) = n-1 17 = n-1 n = 17 + 1 n = 18 The common difference is 1.1 and the number of terms is 18. e) -6, -2(1)/(2), ..., 71 Step 1: Find the common difference d. Convert -2(1)/(2) to a decimal: -2.5. d = a_2 - a_1 = -2.5 - (-6) = -2.5 + 6 = 3.5 Step 2: Find the number of terms n. a_n = a_1 + (n-1)d 71 = -6 + (n-1)3.5 71 + 6 = (n-1)3.5 77 = (n-1)3.5 (77)/(3.5) = n-1 22 = n-1 n = 22 + 1 n = 23 The common difference is 3.5 and the number of terms is 23. That's 2 down. 3 left today — send the next one.