858 CHAPTER 12 Sequences; Induction; the Binomial Theorem The Factorial Symbol Some sequences in mathematics involve a special product called a factorial . Determining a Sequence from a Pattern (a) e e e e , 2 , 3 , 4 , 2 3 4 … = a e n n n (b) 1, 1 3 , 1 9 , 1 27 , … = − b 1 3 n n 1 (c) 1, 3, 5, 7, … = − c n2 1 n (d) 1, 4, 9, 16, 25, … = d n n 2 (e) 1, 1 2 , 1 3 , 1 4 , 1 5 , − − … ( ) ( ) = − − e n 1 1 n n 1 EXAMPLE 4 Now Work PROBLEM 29 TIP When finding a formula for the nth term of a sequence, look for the pattern in the terms given. For example, in part (a), we have e e e e 1 , 2 , 3 , 4 1 2 3 4 n n n n 1 2 3 4 = = = = which makes the formula for the n th term a e n n n = ■ DEFINITION Factorial Symbol If ≥ n 0 is an integer, the factorial symbol n! is defined as follows: ( ) = = = − ⋅ ⋅ ⋅ ⋅ ≥ n n n n • 0! 1 • 1! 1 • ! 1 ...321 if 2 For example, =⋅= =⋅⋅= =⋅⋅⋅= 2! 2 1 2, 3! 3 2 1 6, 4! 4 3 2 1 24,and so on. Table 2 lists the values of n! for ≤ ≤ n 0 6. Because n n n n ! 1 2 3 2 1 ( )( ) = − − ⋅ ⋅ ⋅ ⋅ − ( ) n 1 ! the formula ( ) = ⋅ − n n n ! 1 ! is used to find successive factorials. For example, because = 6! 720, = ⋅ = ⋅ = 7! 7 6! 7 720 5040 and = ⋅ = ⋅ = 8! 8 7! 8 5040 40,320 Now Work PROBLEM 13 2 List the Terms of a Sequence Defined by a Recursive Formula A second way of defining a sequence is to assign a value to the first (or the first few) term(s) and specify the n th term by a formula or equation that involves one or more of the terms preceding it. Such sequences are said to be defined recursively , and the rule or formula is called a recursive formula . n n! 0 1 1 1 2 2 3 6 4 24 5 120 6 720 Table 2 Writing the Terms of a Recursively Defined Sequence Write down the first five terms of the following recursively defined sequence. = = − s s ns 1 n n 1 1 EXAMPLE 5 Exploration Use your graphing utility's factorial key to see how fast factorials increase in value. Find the value of 69!. What happens when you try to find 70!? In fact, 70! is larger than 10100 (a googol), which is the largest number most calculators can display.
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