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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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