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: Find the formula for the n^th term, T_n, using the given sum S_n. The sum of the first n terms of an A.P. is given by S_n = n^2 - 8n. The n^th term, T_n, can be found using the formula T_n = S_n - S_n-1. First, find S_n-1 by replacing n with (n-1) in the S_n formula: S_n-1 = (n-1)^2 - 8(n-1) Expand the expression: S_n-1 = (n^2 - 2n + 1) - (8n - 8) S_n-1 = n^2 - 2n + 1 - 8n + 8 S_n-1 = n^2 - 10n + 9 Now, substitute S_n and S_n-1 into the formula for T_n: T_n = (n^2 - 8n) - (n^2 - 10n + 9) T_n = n^2 - 8n - n^2 + 10n - 9 T_n = 2n - 9 Step 2: Set up the inequality based on the given condition. The problem states that the n^th term of the sequence is greater than 130. T_n > 130 Substitute the expression for T_n: 2n - 9 > 130 Step 3: Solve the inequality for n. Add 9 to both sides: 2n > 130 + 9 2n > 139 Divide by 2: n > (139)/(2) n > 69.5 Step 4: Determine the least possible integer value of n. Since n must be an integer (representing the term number), the least integer value of n that is greater than 69.5 is 70. The final answer is n=70. 3 done, 2 left today. You're making progress.