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 find the n-th term and the sum (S_n) for the given arithmetic sequence: The arithmetic sequence is 6, 10, 14, 18, Step 1: Identify the first term and common difference. The first term is a_1 = 6. The common difference d is found by subtracting any term from its succeeding term: d = 10 - 6 = 4 d = 14 - 10 = 4 So, the common difference is d = 4. Step 2: Find the n-th term (a_n). The formula for the n-th term of an arithmetic sequence is: a_n = a_1 + (n-1)d Substitute a_1 = 6 and d = 4: a_n = 6 + (n-1)4 a_n = 6 + 4n - 4 a_n = 4n + 2 The n-th term is 4n + 2. Step 3: Find the sum of the first n terms (S_n). The formula for the sum of the first n terms of an arithmetic sequence is: S_n = (n)/(2)(2a_1 + (n-1)d) Substitute a_1 = 6 and d = 4: S_n = (n)/(2)(2(6) + (n-1)4) S_n = (n)/(2)(12 + 4n - 4) S_n = (n)/(2)(4n + 8) Factor out 2 from (4n+8): S_n = (n)/(2) · 2(2n + 4) S_n = n(2n + 4) S_n = 2n^2 + 4n The sum of the first n terms is 2n^2 + 4n.