878 CHAPTER 12 Sequences; Induction; the Binomial Theorem THEOREM Sum of the First n Terms of a Geometric Sequence Let an { } be a geometric sequence with first term a1 and common ratio r, where r r 0, 1. ≠ ≠ The sum Sn of the first n terms of an { } is S a a r a r a r a r a r r r 1 1 0, 1 n n k n k n 1 1 1 2 1 1 1 1 1 1 ∑ = + + + + = = ⋅ − − ≠ − = − (3) Proof The sum Sn of the first n terms of a a r n n 1 1 { } { } = − is S a a r a r n n 1 1 1 1 = + + + − (4) Multiply both sides by r to obtain rS a r a r a r n n 1 1 2 1 = + + + (5) Now, subtract (5) from (4). The result is ( ) ( ) − = − − = − S rS a a r r S a r 1 1 n n n n n 1 1 1 Since r 1, ≠ solve for S .n S a r r 1 1 n n 1 = ⋅ − − ■ Finding the Sum of the First n Terms of a Geometric Sequence Find the sum Sn of the first n terms of the sequence 1 2 ; n { } ( ) that is, find 1 2 1 2 1 4 1 8 1 2 k n k n 1 ∑( ) ( ) = + + + + = EXAMPLE 5 Using a Graphing Utility to Find the Sum of a Geometric Sequence Use a graphing utility to find the sum of the first 15 terms of the sequence 1 3 ; n { } ( ) that is, find 1 3 1 3 1 9 1 27 1 3 k k 1 15 15 ∑( ) ( ) = + + + + = EXAMPLE 6 Solution The sequence 1 2 n { } ( ) is a geometric sequence with a 1 2 1 = and r 1 2 . = Use formula (3) to get S 1 2 1 2 1 4 1 8 1 2 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 n k n k n k n k n n n 1 1 1 ∑ ∑ ( ) ( ) ( ) ( ) ( ) ( ) = = + + + + = = ⋅ − − = ⋅ − = − = = − Formula (3); a r 1 2 , 1 2 1 = = Now Work PROBLEM 41 3 Find the Sum of a Geometric Sequence
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