-1, 4, -9, 16, -25,Á1, 14, 19, 116, 125,Á■Answer:-1, 14, -19, 116Answer:a.b.an=12nan=(-1)n2n■Study Tipor is a way to represent an alternating sequence.(-1)n+1(-1)n
1220CHAPTER 12Sequences, Series, and ProbabilityFactorial NotationMany important sequences that arise in mathematics involve terms that are defined withproducts of consecutive positive integers. The products are expressed in factorial notation.If nis a positive integer, then n! (stated as “nfactorial”) is the product of all positiveintegers from ndown to 1.and and 1!=1.0!=1nÚ2n!=n(n-1)(n-2)Á3#2#1FactorialDE FI N ITIONThe values of n! for the first six nonnegative integers areNotice that In general, we can apply the formula Often the brackets are not used, and the notation implies calculating thefactorial and then multiplying that quantity by n.For example, to find 6!, weemploy the relationship and set 6!=6#5!=6#120=720n=6:n!=n(n-1)!(n-1)!n!=n(n-1)!n!=n3(n-1)!4.4!=4#3#2#1=4#3!.5!=5#4#3#2#1=1204!=4#3#2#1=243!=3#2#1=62!=2#1=21!=10!=1Technology TipFind 0!, 1!, 2!, 3!, 4!, and 5!.Scientific calculators:PressDisplay0 ! 11 ! 12 ! 23 ! 64 ! 245 ! 120Graphing calculators:PressDisplay0 MATHPRB4:!ENTER11 MATHPRB4:!ENTER12 MATHPRB4:!ENTER23 MATHPRB4:!ENTER64 MATHPRB4:!ENTER245 MATHPRB4:!ENTER120EXAMPLE 3Finding the Terms of a Sequence Involving FactorialsFind the first four terms of the sequence, given the general term .Solution:Find the first term,Find the second term,Find the third term,Find the fourth term,The first four terms of the sequence are.x, x22, x36, x424a4=x44!=x44#3#2#1=x424n=4.a3=x33!=x33#2#1=x36n=3.a2=x22!=x22#1=x22n=2.a1=x11!=xn=1.an=xnn!Parts (b) in both Example 1 and Example 2 are called alternatingsequences because theterms alternate signs (positive and negative). If the odd terms,are negative and the even terms,are positive, we include in the general term. If theopposite is true, and the odd terms are positive and the even terms are negative, we includein the general term.(-1)n+1(-1)na2, a4, a6,Á,a1, a3, a5,Á
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- Summer '17
- juan alberto
- Arithmetic progression, Geometric progression, Conic section, general term, graphing utility