10-2 Arithmetic Sequences and Series
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1 Determine the common difference, and find the next four terms of each arithmetic sequence , 17, 14, = = 3 The common difference is 3. Add 3 to the third term to find the fourth term, and so on ( 3) = ( 3) = ( 3) = ( 3) = 2 Therefore, the next four terms are 11, 8, 5, and , 108, 99, = = 9 The common difference is 9. Add 9 to the third term to find the fourth term, and so on = = = = 63 Therefore, the next four terms are 90, 81, 72, and , 1, 5, 1 ( 3) = = 4 The common difference is 4. Add 4 to the third term to find the fourth term, and so on = = = = 21 Therefore, the next four terms are 9, 13, 17, and 21 esolutions Manual - Powered by Cognero Page 1
2 7. 4.5, 9.5, 14.5, 9.5 ( 4.5) = ( 9.5) = 5 The common difference is 5. Add 5 to the third term to find the fourth term, and so on = = = = 34.5 Therefore, the next four terms are 19.5, 24.5, 29.5, and MARCHING BAND A marching band begins its performance in a pyramid formation. The first row has 1 band member, the second row has 3 band members, the third row has 5 band members, and so on. a. Find the number of band members in the 8th row. b. Write an explicit formula and a recursive formula for finding the number of band members in the nth row. a. The first row has 1 band member, the second row has 3 band members, and the third row has 5 band members, so a 1 = 1, a 2 = 3, and a 3 = 5. Find the common difference. 3 1 = = 2 The common difference is 2. Add 2 to the third term to find the fourth term, and so on to find the eighth term = = = = = 15 So, there are 15 band members in the 8th row. b. For an explicit formula, substitute a 1 = 1 and d = 2 in the formula for the nth term of an arithmetic sequence. For the recursive formula, state the first term a 1 and then indicate that the next term is the sum of the previous term a n 1 and d. a 1 = 1, a n = a n esolutions Manual - Powered by Cognero Page 2
3 Find both an explicit formula and a recursive formula for the nth term of each arithmetic sequence , 5, 16, 5 ( 6) = = 11 For an explicit formula, substitute a 1 = 6 and d = 11 in the formula for the nth term of an arithmetic sequence. For the recursive formula, state the first term a 1 and then indicate that the next term is the sum of the previous term a n 1 and d. a 1 = 6, a n = a n , 19, 34, 19 4 = = 15 For an explicit formula, substitute a 1 = 4 and d = 15 in the formula for the nth term of an arithmetic sequence. For the recursive formula, state the first term a 1 and then indicate that the next term is the sum of the previous term a n 1 and d. a 1 = 4, a n = a n esolutions Manual - Powered by Cognero Page 3
4 15. 7, 3.5, 14, = ( 3.5) = 10.5 For an explicit formula, substitute a 1 = 7 and d = 10.5 in the formula for the nth term of an arithmetic sequence. For the recursive formula, state the first term a 1 and then indicate that the next term is the sum of the previous term a n 1 and d. a 1 = 7, a n = a n , 37, 73, 37 1 = = 36 For an explicit formula, substitute a 1 = 1 and d = 36 in the formula for the nth term of an arithmetic sequence. For the recursive formula, state the first term a 1 and then indicate that the next term is the sum of the previous term a n 1 and d. a 1 = 1, a n = a n Find the specified value for the arithmetic sequence with the given characteristics. 19. Find d for 24, 31, 38, Find the difference between two pairs of consecutive terms. 31 ( 24) = 7 38 ( 31) = 7 Therefore, d = 7. esolutions Manual - Powered by Cognero Page 4
5 21. If a 1 = 47 and d = 5, find a 12. Substitute a 1 = 47, n = 12, and d = 5 into the formula for the nth term of an arithmetic sequence. 23. Find a 6 for 84, 5, 74, 5 84 = = 79 Substitute a 1 = 84, n = 6, and d = 79 into the formula for the nth term of an arithmetic sequence. 25. If a 35 = 63 and a 1 = 39, find d. Substitute a 1 = 39, n = 35, and a 35 = 63 into the formula for the nth term of an arithmetic sequence. esolutions Manual - Powered by Cognero Page 5
6 Find the indicated arithmetic means for each set of nonconsecutive terms means; 19 and 5 The sequence will resemble 19,?,?,?, 5. Note that 5 is the fifth term of the sequence or a 5. First, find the common difference using a 5 = 5, a 1 = 19, and n = 5. Next, determine the arithmetic means by using d = ( 6) = ( 6) = ( 6) = 1 Therefore, a sequence with three arithmetic means between 19 and 5 is 19, 13, 7, 1, means; 3 and 88 The sequence will resemble 3,?,?,?,?, 88. Note that 88 is the sixth term of the sequence or a 6. First, find the common difference using a 6 = 88, a 1 = 3, and n = 6. Next, determine the arithmetic means by using d = = = = = 71 Therefore, a sequence with four arithmetic means between 3 and 88 is 3, 20, 37, 54, 71, 88. esolutions Manual - Powered by Cognero Page 6
7 31. 7 means; 4.5 and 7.5 The sequence will resemble 4.5,?,?,?,?,?,?,?, 7.5. Note that 7.5 is the ninth term of the sequence or a 9. First, find the common difference using a 9 = 7.5, a 1 = 4.5, and n = 9. Next, determine the arithmetic means by using d = = = = = = = = 6 Therefore, a sequence with seven arithmetic means between 4.5 and 7.5 is 3, 1.5, 0, 1.5, 3, 4.5, 6. esolutions Manual - Powered by Cognero Page 7
8 Find a quadratic model for each sequence , 19, 28, 39, 52, 67, The nth term can be represented by a quadratic equation of the form a n = an 2 + bn + c. Substitute values for a n and n into the equation. This yields a system of linear equations in three variables. You can use a graphing calculator to solve for a, b, and c. First, write the augmented matrix that corresponds to the system. Next, enter the matrix into your graphing calculator, and use the rref( feature under the 2nd [MATRIX] MATH menu to find the solution. Therefore, a = 1, b = 4, and c = 7. Substituting these values in the equation for a n, the model for the sequence is a n = n 2 + 4n + 7c. esolutions Manual - Powered by Cognero Page 8
9 35. 8, 3, 6, 19, 36, 57, The nth term can be represented by a quadratic equation of the form a n = an 2 + bn + c. Substitute values for a n and n into the equation. This yields a system of linear equations in three variables. You can use a graphing calculator to solve for a, b, and c. First, write the augmented matrix that corresponds to the system. Next, enter the matrix into your graphing calculator, and use the rref( feature under the 2nd [MATRIX] MATH menu to find the solution. Therefore, a = 2, b = 1, and c = 9. Substituting these values in the equation for a n, the model for the sequence is a n = 2n 2 + n + 9. esolutions Manual - Powered by Cognero Page 9
10 37. 6, 2, 12, 24, 38, 54, The nth term can be represented by a quadratic equation of the form a n = an 2 + bn + c. Substitute values for a n and n into the equation. This yields a system of linear equations in three variables. You can use a graphing calculator to solve for a, b, and c. First, write the augmented matrix that corresponds to the system. Next, enter the matrix into your graphing calculator, and use the rref( feature under the 2nd [MATRIX] MATH menu to find the solution. Therefore, a = 1, b = 5, and c = 12. Substituting these values in the equation for a n, the model for the sequence is a n = n 2 5n esolutions Manual - Powered by Cognero Page 10
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