SECTION 12.3 Geometric Sequences; Geometric Series 877 2 Find a Formula for a Geometric Sequence Suppose that a1 is the first term of a geometric sequence with common ratio r 0. ≠ We seek a formula for the n th term, a .n To see the pattern, consider the first few terms: a a a r a ra a r a ra r a r a r a ra r a r a r a ra r a r a r a ra r a r a r 1 n n n n 1 1 1 0 2 1 1 1 3 2 1 1 2 4 3 1 2 1 3 5 4 1 3 1 4 1 1 2 1 1 ( ) ( ) ( ) ( ) = ⋅ = = = = = = = = = = = = = = = − − − The terms of a geometric sequence with first term a1 and common ratio r follow the pattern … a a r a r a r , , , , 1 1 1 2 1 3 THEOREM n th Term of a Geometric Sequence For a geometric sequence { } an whose first term is a1 and whose common ratio is r, the n th term is determined by the formula a a r r 0 n n 1 1 = ≠ − (2) Finding a Particular Term of a Geometric Sequence (a) Find the n th term of the geometric sequence: … 10, 9, 81 10 , 729 100 , (b) Find the 9th term of the sequence. (c) Find a recursive formula for the sequence. Solution EXAMPLE 4 (a) The first term of the geometric sequence is a 10. 1 = The common ratio r a a n n 1 = − is the ratio of any two consecutive terms. So, r a a 9 10 . 2 1 = = Then, by formula (2), the n th term of the geometric sequence is a 10 9 10 n n 1 ( ) = − a a r a r ; 10, 9 10 n n 1 1 1 = = = − (b) The 9th term is a 10 9 10 10 9 10 4.3046721 9 9 1 8 ( ) ( ) = = = − (c) The first term in the sequence is 10, and the common ratio is r 9 10 . = Using formula (1), the recursive formula is a a a 10, 9 10 . n n 1 1 = = − Exploration Use a graphing utility to find the ninth term of the sequence in Example 4. Use it to find the 20th and 50th terms. Now use a graphing utility to graph the recursive formula found in Example 4(c). Conclude that the graph of the recursive formula behaves like the graph of an exponential function. How is r, the common ratio, related to a, the base of the exponential function = y a ? x Now Work PROBLEMS 19, 27, AND 35
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