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
Hey Ngangwa, good to see you again. Step 1: Write the given formula for the sum of the first n terms, S_n. S_n = 3n^2 + n Step 2: Find the formula for the sum of the first (n-1) terms, S_n-1, by replacing n with (n-1) in the S_n formula. S_n-1 = 3(n-1)^2 + (n-1) Expand the term (n-1)^2: S_n-1 = 3(n^2 - 2n + 1) + (n-1) Distribute the 3: S_n-1 = 3n^2 - 6n + 3 + n - 1 Combine like terms: S_n-1 = 3n^2 - 5n + 2 Step 3: Use the relationship T_n = S_n - S_n-1 to find the n-th term, T_n. T_n = (3n^2 + n) - (3n^2 - 5n + 2) Step 4: Simplify the expression. T_n = 3n^2 + n - 3n^2 + 5n - 2 Combine the n^2 terms: 3n^2 - 3n^2 = 0. Combine the n terms: n + 5n = 6n. The constant term is -2. T_n = 6n - 2 The n-th term is: T_n = 6n - 2 Send me the next one 📸