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
To find the n^th term and the three terms of the arithmetic sequence, we first need to determine the common difference (d) and the first term (a_1). Given: a_4 = 93 a_8 = 65 Step 1: Find the common difference (d). In an arithmetic sequence, the n^th term can be expressed as a_n = a_k + (n-k)d. Using a_8 and a_4: a_8 = a_4 + (8-4)d a_8 = a_4 + 4d Substitute the given values: 65 = 93 + 4d Subtract 93 from both sides: 65 - 93 = 4d -28 = 4d Divide by 4: d = (-28)/(4) d = -7 Step 2: Find the first term (a_1). The general formula for the n^th term of an arithmetic sequence is a_n = a_1 + (n-1)d. Using a_4 = 93 and d = -7: a_4 = a_1 + (4-1)d 93 = a_1 + 3(-7) 93 = a_1 - 21 Add 21 to both sides: a_1 = 93 + 21 a_1 = 114 Step 3: Find the n^th term (a_n). Using the general formula a_n = a_1 + (n-1)d with a_1 = 114 and d = -7: a_n = 114 + (n-1)(-7) a_n = 114 - 7n + 7 a_n = 121 - 7n Step 4: Find the first three terms of the sequence. The first term is a_1 = 114. The second term is a_2 = a_1 + d = 114 + (-7) = 107. The third term is a_3 = a_2 + d = 107 + (-7) = 100. The n^th term of the arithmetic sequence is 121 - 7n. The first three terms of the sequence are 114, 107, 100.