870 CHAPTER 12 Sequences; Induction; the Binomial Theorem 2 Find a Formula for an Arithmetic Sequence Suppose that a is the first term of an arithmetic sequence whose common difference is d. We seek a formula for the n th term, a .n To see the pattern, consider the first few terms. ( ) ( ) ( ) [ ] ( ) ( ) = = + = + ⋅ = + = + + = + ⋅ = + = + ⋅ + = + ⋅ = + = + ⋅ + = + ⋅ = + = + − + = + − − a a a a d a d a a d a d d a d a a d a d d a d a a d a d d a d a a d a n d d a n d 1 2 2 3 3 4 2 1 n n 1 2 1 1 3 2 1 1 4 3 1 1 5 4 1 1 1 1 1 The terms of an arithmetic sequence with first term a1 and common difference d follow the pattern + + + a a d a d a d , , 2 , 3 , . . . 1 1 1 1 THEOREM n th Term of an Arithmetic Sequence For an arithmetic sequence { } an whose first term is a1 and whose common difference is d, the n th term is determined by the formula ( ) = + − a a n d n 1 a positive integer n 1 (2) Determining Whether a Sequence Is Arithmetic Show that the following sequence is arithmetic. Find the first term and the common difference. { } { } = + s n3 5 n Solution EXAMPLE 2 The first term is = ⋅ + = s 3 1 5 8. 1 The n th term and the ( ) −n 1 st term of the sequence { } sn are ( ) = + = − + = + − s n s n n 3 5 and 3 1 5 3 2 n n 1 Their difference d is ( ) ( ) = − = + − + = − = − d s s n n 3 5 3 2 5 2 3 n n 1 Since the difference of any two successive terms is 3, { } sn is an arithmetic sequence. The common difference is = d 3. Determining Whether a Sequence Is Arithmetic Show that the sequence { } { } = − t n 4 n is arithmetic. Find the first term and the common difference. Solution EXAMPLE 3 The first term is = − = t 4 1 3. 1 The n th term and the ( ) −n 1 st term are ( ) = − = − − = − − t n t n n 4 and 4 1 5 n n 1 Their difference d is ( ) ( ) =− =−−−=−=− − d t t n n 4 5 4 5 1 n n 1 Since the difference of any two successive terms is −1, { } tn is an arithmetic sequence. The common difference is = − d 1. Now Work PROBLEM 9
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