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9.1 Sequences Objec(ves: • Define sequence • Arithme(c and geometric sequences • Graph a sequence and determine the limit. Assignment: AB Book pg. 604 #’s 1-‐33 odd, 35-‐44 all Sequences A sequence {an } is a list of numbers written in explicit order. an = {a1 ,a2 ,a3 ,...,an ,...} a1 ! First term a2 ! 2 nd term a3 ! 3rd term an ! n th term Sequences Any real-valued function with domain a subset of the set of positive integers is considered a sequence. If the domain is finite, then the sequence is a finite sequence. 1 Example: Find the first 4 terms and the 100th term of the sequence{an }where n an = ( !1) n2 + 1 This sequence is an explicit sequence because an is defined in terms of n. Another way to define a sequence is recursively by giving a formula for an in terms of the previous term. Example: Find the first 4 terms and the 8th term for the sequence defined recursively by the conditions b1 = 4 bn = bn!1 + 2 for all n " 2 There are many types of sequences but two particular types of sequences are dominant in mathematical applications. • Arithmetic Sequences- pairs of successive terms all have common differences. • Geometric Sequences- pairs of successive terms all have common ratios. 2 Arithmetic Sequences A sequence {an }is a arithmetic sequence if it can be written in the form {a, a + d, a + 2d,... a + (n ! 1)d,...} for some constant d. The number d is the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d: an = an!1 + d, for all n " 2 Example: For the sequence, find the common difference, the ninth term, a recursive rule for the nth term and an explicit rule for the nth term. -5, -2, 1, 4, 7, … Common Difference: Ninth Term: Recursive Rule: Explicit Rule: Geometric Sequence A sequence {an }is a geometric sequence if it can be written in the form {a, a ! r, a ! r ,... a ! r 2 n"1 } ,... for some nonzero constant r. The number r is the common ratio. Each term in an geometric sequence can be obtained recursively from its preceding term by multiplying by r: an = an!1 " r for all n # 2 3 Example: For the sequence, find the common ratio, the 10th term, a recursive rule for the nth term and an explicit rule for the nth term. 1, -2, 4, -8, 16, … Common Ratio: 10th Term: Recursive Rule: Explicit Rule: Example: The second and fifth term of geometric sequence are 6 and -48 respectively. Find the first term, common ratio, and an explicit rule for the nth term. The main focus of this chapter concerns sequences whose terms approach limi(ng values. Such sequences are said to converge. 1 1 1 1 1 1 , , , , ,... n 2 4 8 16 32 2 This sequence converges to 0. 4 Limit Let L be a real number. The sequence {an }has limit L as n approaches ∞ if, given any positive number ε, there is a positive number M such that for all n > M we have an ! L < ! We write lim an = L and say that the sequence n!" converges to L. Sequences that do not have limits diverge. There are different ways in which a sequences can fail to have a limit. One way is that the terms of the sequence increase without bound or decrease without bound. These cases are wriRen symbolically as follows: Properties of Limits If L and M are real numbers and lim an = L and n!" lim bn = M , then n!" ( an ± bn ) = L ± M 1. Sum and Difference Rule lim n!" ( anbn ) = L # M 2. Product Rule lim n!" ( c # an ) = c # L 3. Constant Multiple Rule lim n!" 4. Quotient Rule #a & L lim % n ( = , M ) 0 n!" $ b ' M n 5 Example: Determine whether the sequence converges or diverges. If it converges, find its limit. an = 2n ! 1 n 6