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Stitz-Zeager_College_Algebra_e-book

# 94 the binomial theorem 583 songs matters it turns out

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Unformatted text preview: ribe the factorial sequence as: 0! = 1 and n! = n(n − 1)! for n ≥ 1. After 0! = 1 the next four terms, written out in detail, are 1! = 1 · 0! = 1 · 1 = 1, 2! = 2 · 1! = 2 · 1 = 2, 3! = 3 · 2! = 3 · 2 · 1 = 6 and 4! = 4 · 3! = 4 · 3 · 2 · 1 = 24. From this, we see a more informal way of computing n!, which is n! = n · (n − 1) · (n − 2) · · · 2 · 1 with 0! = 1 as a special case. (We will study factorials in greater detail in Section 9.4.) The world famous Fibonacci Numbers are deﬁned recursively and are explored in the exercises. While none of the sequences worked out to be the sequence in (1), they do give us some insight into what kinds of patterns to look for. Two patterns in particular are given in the next deﬁnition. Definition 9.2. Arithmetic and Geometric Sequences: Suppose {an }∞ k is a sequencea n= • If there is a number d so that an+1 = an + d for all n ≥ k , then {an }∞ k is called an n= arithmetic sequence. The number d is called the common diﬀerence. • If there is a number r so that an+1 = ran for all n ≥ k , the...
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