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.
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Here's how to solve this problem: Given an arithmetic sequence with the first term a = 6 and the 20th term T_20 = 82. The formula for the n-th term of an arithmetic sequence is T_n = a + (n-1)d. a) Find the common difference d. Step 1: Substitute the given values into the formula for T_20. T_20 = a + (20-1)d 82 = 6 + 19d Step 2: Solve for d. 82 - 6 = 19d 76 = 19d d = (76)/(19) d = 4 Answer: d = 4 b) Find the 100th term. Step 1: Use the first term a=6 and the common difference d=4 to find T_100. T_100 = a + (100-1)d T_100 = 6 + (99)4 Step 2: Calculate the value. T_100 = 6 + 396 T_100 = 402 Answer: 402 c) Which term of this sequence equals 226? Step 1: Set T_n = 226 in the formula T_n = a + (n-1)d. 226 = 6 + (n-1)4 Step 2: Solve for n. 226 - 6 = (n-1)4 220 = (n-1)4 (220)/(4) = n-1 55 = n-1 n = 55 + 1 n = 56 Answer: The 56th term Last free one today — make it count tomorrow, or type /upgrade for unlimited.