1055 11.2 Arithmetic Sequences and Series The sequence consists of the points 11, 3.52, 12, 32, 13, 2.52, 14, 22, 15, 1.52, 16, 12. Thus, the domain of the given sequence is 51, 2, 3, 4, 5, 66, and the range is 53.5, 3, 2.5, 2, 1.5, 16. S Now Try Exercise 39. Sum of the First n Terms of an Arithmetic Sequence If an arithmetic sequence has first term a1 and common difference d, then the sum Sn of the first n terms is given by the following. Sn = n 2 1 a1 +an2, or Sn = n 2 3 2 a1 + 1n −12d4 The first formula is used when the first and last terms are known; otherwise, the second formula is used. Arithmetic Series An arithmetic series is the sum of the terms of an arithmetic sequence. To illustrate, suppose that a person borrows $3000 and agrees to pay $100 per month plus interest of 1% per month on the unpaid balance until the loan is paid off. The first month, $100 is paid to reduce the loan, plus interest of 10.0123000 = 30 dollars. The second month, another $100 is paid toward the loan, plus interest of 10.0122900 = 29 dollars. The loan is reduced by $100 each month. Interest payments decrease by 10.012100 = 1 dollar each month, forming the arithmetic sequence 30, 29, 28, c , 3, 2, 1. The total amount of interest paid is given by the sum of the terms of this sequence. Now we develop a formula to find this sum without adding all 30 numbers directly. Because the sequence is arithmetic, we write the sum of the first n terms as follows. Sn = a1 + 1a1 + d2 + 1a1 + 2d2 + g+ 3a1 + 1n - 12d4 Now we write the same sum in reverse order, beginning with an and subtracting d. Sn = an + 1an - d2 + 1an - 2d2 + g+ 3an - 1n - 12d4 Adding the respective sides of these two equations term by term, we obtain the following. Sn + Sn = 1a1 + an2 + 1a1 + an2 + g+ 1a1 + an2 2Sn = n1a1 + an2 There are n terms of a1 + an on the right. Sn = n 2 1a1 + an2 Solve for Sn. Using the formula an = a1 + 1n - 12d, we can also write this result as follows. Sn = n 2 3a1 + a1 + 1n - 12d4 Let an = a1 + 1n - 12d. or Sn = n 2 32a1 + 1n - 12d4 Alternative formula for the sum of the first n terms
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