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 the problem for the arithmetic sequence 7, 12, 17, 22, ...: a) Write down the first term a and the common difference d. Step 1: Identify the first term. The first term a is the first number in the sequence. a = 7 Step 2: Calculate the common difference. The common difference d is found by subtracting any term from its succeeding term. d = 12 - 7 = 5 Answer: a = 7, d = 5 b) Find the 20th term. Step 1: Use the formula for the n-th term of an arithmetic sequence, T_n = a + (n-1)d. Here, a = 7, d = 5, and n = 20. Step 2: Substitute the values into the formula. T_20 = 7 + (20-1)5 Step 3: Calculate the value. T_20 = 7 + (19)5 T_20 = 7 + 95 T_20 = 102 Answer: 102 c) Find the 50th term. Step 1: Use the formula for the n-th term of an arithmetic sequence, T_n = a + (n-1)d. Here, a = 7, d = 5, and n = 50. Step 2: Substitute the values into the formula. T_50 = 7 + (50-1)5 Step 3: Calculate the value. T_50 = 7 + (49)5 T_50 = 7 + 245 T_50 = 252 Answer: 252 d) Is 102 a term in this sequence? If yes, which term is it? Step 1: Set T_n = 102 in the formula T_n = a + (n-1)d. 102 = 7 + (n-1)5 Step 2: Solve for n. 102 - 7 = (n-1)5 95 = (n-1)5 (95)/(5) = n-1 19 = n-1 n = 19 + 1 n = 20 Step 3: Determine if n is a positive integer. Since n=20 is a positive integer, 102 is a term in the sequence. Answer: Yes, 102 is the 20th term. That's 2 down. 3 left today — send the next one.