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, a_n. The sum of the first n terms of an arithmetic progression is given by S_n = n^2 - 8n. The n^th term, a_n, can be found using the relationship a_n = S_n - S_n-1. First, find S_n-1 by substituting (n-1) for n in the formula for S_n: S_n-1 = (n-1)^2 - 8(n-1) 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 2: Calculate a_n. Now, substitute S_n and S_n-1 into the formula for a_n: a_n = S_n - 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 3: Set up the inequality. 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 4: 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 5: Determine the least possible integer value of n. Since n must be an integer (as it represents the term number) and n must be greater than 69.5, the least possible integer value for n is 70. The least possible value of n is 70. That's 2 down. 3 left today — send the next one.