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-1)^th sum, S_n-1. Given the sum of the first n terms of an arithmetic progression (A.P.) is S_n = n^2 - 8n. To find S_n-1, substitute (n-1) for n: S_n-1 = (n-1)^2 - 8(n-1) Step 2: Expand and simplify S_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 Step 3: Find the formula for the n^th term, a_n. The n^th term of an A.P. can be found using the formula a_n = S_n - S_n-1. Substitute the expressions for S_n and S_n-1: a_n = (n^2 - 8n) - (n^2 - 10n + 9) a_n = n^2 - 8n - n^2 + 10n - 9 a_n = 2n - 9 Step 4: Set up the inequality based on the problem statement. The problem states that the n^th term of the sequence is greater than 130. a_n > 130 Substitute the expression for a_n: 2n - 9 > 130 Step 5: 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 6: Determine the least possible integer value of n. Since n represents the term number, it must be an integer. The least integer value of n that is greater than 69.5 is 70. The least possible value of n is 70. Send me the next one 📸