276 CHAPTER 5 Number Theory and the Real Number System When discussing a sequence, we often represent the first term as a1 (read “ a sub 1”), the second term as a ,2 the fifteenth term as a , 15 and so on. We use the notation an to represent the general or nth term of a sequence. Thus, a sequence may be symbolized as … … a a a a a , , , , , ,n 1 2 3 4 For example, in the sequence … 2, 5, 8, 11, 14, , we have = = = = = … a a a a a 2, 5, 8, 11, 14, . 1 2 3 4 5 The following formula can be used to determine the nth term of an arithmetic sequence. Profile in Mathematics Carl Friedrich Gauss Carl Friedrich Gauss (1777– 1855), often called the “Prince of Mathematicians,” made significant contributions to the fields of algebra, geometry, and number theory. Gauss was only 22 years old when he proved the fundamental theorem of algebra for his doctoral dissertation. When Gauss was only 10, his mathematics teacher gave him the problem of determining the sum of the first 100 natural numbers, thinking that this task would keep Gauss busy for awhile. Gauss recognized a pattern in the sequence of numbers when he considered the sum of the following numbers. 1 2+ 3+ 99 100 100 9998 2 1 101 101 101 101 101 + + + + + + + + + + + + + He had the required answer in no time at all. When he added, he had one hundred 101’s. Therefore, the sum of the first 100 natural numbers is (100)(101) 5050. 1 2 = General or th n Term of an Arithmetic Sequence For an arithmetic sequence with first term a1 and common difference d, the general or nth term can be determined using the following formula. a a n d ( 1) n 1 = + − Example 3 Determining the Tenth Term of an Arithmetic Sequence Determine the tenth term of the arithmetic sequence whose first term is 3− and whose common difference is 5. Solution To determine the tenth term, or a , 10 replace n in the formula with 10, a1 with 3, − and d with 5. a a n d a ( 1) 3 (10 1)5 3 (9)5 3 45 42 n 1 10 = + − = − + − = − + = − + = The tenth term is 42. As a check, we have listed the first 10 terms of the sequence. 3, 2, 7, 12, 17, 22, 27, 32, 37, 42. − 7 Now try Exercise 13 Example 4 Determining an Expression for the nth Term Write an expression for the general or nth term, a ,n for the sequence 1, 6, 11, 16, .… Solution In this sequence, the first term, a ,1 is 1 and the common difference, d, is 5. We substitute these values into a a n d ( 1) n 1 = + − to obtain an expression for the nth term, a .n a a n d n n n ( 1) 1 ( 1)5 1 5 5 5 4 n 1 = + − = + − = + − = − The expression is a n5 4. n = − Note that when n 1, = the first term is 5(1) 4 1. − = When n 2, = the second term is − = 5(2) 4 6, and so on. 7 Now try Exercise 19
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