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
Step 1: Set up equations for the given terms. The formula for the n-th term of an arithmetic progression is a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. For the third term (a_3 = 9): a_3 = a_1 + (3-1)d 9 = a_1 + 2d (*) For the ninth term (a_9 = 27): a_9 = a_1 + (9-1)d 27 = a_1 + 8d (**) Step 2: Solve the system of equations to find a_1 and d. Subtract equation () from equation (*): (a_1 + 8d) - (a_1 + 2d) = 27 - 9 6d = 18 d = (18)/(6) d = 3 Substitute d=3 into equation (*): 9 = a_1 + 2(3) 9 = a_1 + 6 a_1 = 9 - 6 a_1 = 3 Step 3: Find the fifth term (a_5). Using the formula a_n = a_1 + (n-1)d with n=5, a_1=3, and d=3: a_5 = a_1 + (5-1)d a_5 = 3 + 4(3) a_5 = 3 + 12 a_5 = 15 The fifth term is 15. 3 done, 2 left today. You're making progress.