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
Here are the solutions to the problems: 1. 16th term of the arithmetic sequence 5,12,19,26, ... ... ... is Step 1: Identify the first term (a) and the common difference (d). The first term is a = 5. The common difference is d = 12 - 5 = 7. Step 2: Use the formula for the n^th term of an arithmetic sequence, a_n = a + (n-1)d. For the 16th term (n=16): a_16 = 5 + (16-1) × 7 a_16 = 5 + 15 × 7 a_16 = 5 + 105 a_16 = 110 The correct option is C) 110. 110 2. What is the difference between the 20th term and 15th term of the arithmetic sequence whose common difference 5? Step 1: Identify the common difference (d). The common difference is d = 5. Step 2: Use the formula for the n^th term, a_n = a + (n-1)d, to find the difference. The difference between the 20th term and the 15th term is a_20 - a_15. a_20 - a_15 = (a + (20-1)d) - (a + (15-1)d) a_20 - a_15 = (a + 19d) - (a + 14d) a_20 - a_15 = 19d - 14d a_20 - a_15 = 5d Step 3: Substitute the value of d. a_20 - a_15 = 5 × 5 a_20 - a_15 = 25 The correct option is C) 25. 25 3. If the 16th term of an arithmetic sequence is 47, what is the sum of the 31st terms? Step 1: Write the given information using the formula for the n^th term. The 16th term is a_16 = a + (16-1)d = a + 15d. Given a_16 = 47, so a + 15d = 47. Step 2: Use the formula for the sum of the first n terms of an arithmetic sequence, S_n = (n)/(2)(2a + (n-1)d). We need to find the sum of the first 31 terms, S_31. S_31 = (31)/(2)(2a + (31-1)d) S_31 = (31)/(2)(2a + 30d) S_31 = (31)/(2) × 2(a + 15d) S_31 = 31(a + 15d) Step 3: Substitute the value of a + 15d from Step 1. S_31 = 31 × 47 S_31 = 1457 The correct option is A) 1457. 1457 4. (i) Write the sequence of natural numbers which leaves a remainder 1 on division by 4. (ii) Is this an arithmetic sequence? Why? i) Step 1: List the natural numbers that leave a remainder of 1 when divided by 4. These numbers are 1, 5, 9, 13, 17, ii) Step 2: Determine if the sequence is an arithmetic sequence and explain why. Yes, this is an arithmetic sequence. The difference between consecutive terms is constant: 5 - 1 = 4 9 - 5 = 4 13 - 9 = 4 Since the common difference (d=4) is constant, it is an arithmetic sequence. 5. The Sum of terms from 4th to 10th of an arithmetic sequence is 84, i) What is the 7th term? ii) What is the sum of 6th term and 8th term. i) Step 1: Identify the terms and their sum. The terms are a_4, a_5, a_6, a_7, a_8, a_9, a_10. There are 10 - 4 + 1 = 7 terms. The sum of these 7 terms is 84. Step 2: Use the property that for an arithmetic sequence, the sum of an odd number of terms is the number of terms multiplied by the middle term. The middle term of these 7 terms is the $(7+1)/(2) = 4^