Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: n {an }∞ k is called an geometric n= sequence. The number r is called the common ratio. a Note that we have adjusted for the fact that not all ‘sequences’ begin at n = 1. Both arithmetic and geometric sequences are defined in terms of recursion equations. In English, an arithmetic sequence is one in which we proceed from one term to the next by always adding the fixed number d. The name ‘common difference’ comes from a slight rewrite of the recursion equation from an+1 = an + d to an+1 − an = d. Analogously, a geometric sequence is one in which we proceed from one term to the next by always multiplying by the same fixed number r. If r = 0, we can rearrange the recursion equation to get an+1 = r, hence the name ‘common ratio.’ Some an sequences are arithmetic, some are geometric and some are neither as the next example illustrates.3 Example 9.1.2. Determine if the following sequences are arithmetic, geometric or neither. If arithmetic, find the common difference d; if geometric, find the common ratio r. 1. an = 5n−1 ,n≥1 3n 3. {2n − 1}∞ n=1 2. bk = (−1)k...
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