872 CHAPTER 12 Sequences; Induction; the Binomial Theorem Now Work PROBLEM 39 Solution The sequence is an arithmetic sequence with first term = a 8 1 and common difference = − = d 11 8 3. To find the sum of the first n terms, use formula (3). ( ) [ ] ( ) = ⋅ + − ⋅ = + S n n n n 2 2 8 1 3 2 3 13 n ↑ S n a n d 2 2 1 n 1 ( ) [ ] = + − Now Work PROBLEM 43 Finding the Sum of an Arithmetic Sequence Find the sum: + + + + + 60 64 68 72 120 Solution EXAMPLE 7 This is the sum Sn of an arithmetic sequence { } an whose first term is = a 60 1 and whose common difference is = d 4. The nth term is = a 120. n Use formula (2) to find n. ( ) ( ) ( ) = + − = + − ⋅ = − = − = a a n d n n n n 1 120 60 1 4 60 4 1 15 1 16 n 1 Formula (2) a a d 120, 60, 4 n 1 = = = Simplify. Simplify. Solve for n. Now use formula (4) to find the sum S . 16 ( ) + + + + = = + = S 60 64 68 120 16 2 60 120 1440 16 ( ) = + S n a a 2 n n 1 There are two ways to find the sum of the first n terms of an arithmetic sequence. Formula (3) uses the first term and common difference, and formula (4) uses the first term and the nth term. Use whichever form is easier. Proof ( ) ( ) ( ) [ ] ( ) [ ] [ ] ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) = + + + + = + + + + + + + − = + + + + + + + − = + + + + − = + ⋅ − = + − = + − = + + − = + S a a a a a a d a d a n d a a a d d n d na d n na d n n na n n d n a n d n a a n d n a a 2 1 2 1 1 2 1 1 2 2 1 2 2 1 2 1 2 n n n 1 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n terms Sum of first n terms Formula (2) Rearrange terms. ∑ ( ) = − = − k n n 1 2 k n 1 1 Factor out n 2 ; this is formula (3). ( ) = + − a a n d 1 ; n 1 this is formula (4). ■ Finding the Sum of an Arithmetic Sequence Find the sum Sn of the first n terms of the arithmetic sequence + + + + 8 11 14 17 EXAMPLE 6
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