876 CHAPTER 12 Sequences; Induction; the Binomial Theorem Determining Whether a Sequence Is Geometric Show that the following sequence is geometric. t 3 4 n n { } { } = ⋅ Find the first term and the common ratio. Solution EXAMPLE 3 The first term is t 3 4 12. 1 1 = ⋅ = The n th term and the n 1 st ( ) − term are t t 3 4 and 3 4 n n n n 1 1 = ⋅ = ⋅ − − Their ratio is t t 3 4 3 4 4 4 n n n n n n 1 1 1 = ⋅ ⋅ = = ( ) − − − − The sequence, { } t , n is a geometric sequence with common ratio 4. Now Work PROBLEM 11 *Sometimes called a geometric progression . Determining Whether a Sequence Is Geometric The sequence … 2, 6, 18, 54, 162, is geometric because the ratio of successive terms is 3; 6 2 18 6 54 18 3 . ( ) = = = = The first term is a 2, 1 = and the common ratio is 3. EXAMPLE 1 Solution The first term of the sequence is s 2 1 2 . 1 1 = = − The n th term and the n 1 st ( ) − term of the sequence { } sn are s s 2 and 2 n n n n 1 1 = = ( ) − − − − Their ratio is s s 2 2 2 2 1 2 n n n n n n 1 1 1 1 = = = = ( ) ( ) − − − − − + − − Because the ratio of successive terms is the nonzero constant 1 2 , the sequence { } sn is geometric and the common ratio is 1 2 . Determining Whether a Sequence Is Geometric Show that the following sequence is geometric. s 2 n n { } { } = − Find the first term and the common ratio. EXAMPLE 2 DEFINITION Geometric Sequence A geometric sequence * is defined recursively as a a a a r , , n n 1 1 = = − or as a a a ra n n 1 1 = = − (1) where a a 1 = and r 0 ≠ are real numbers.The number a1 is the first term, and the nonzero number r is called the common ratio .
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