Geometric Sequences and Series. To recognize, write, and use geometric sequences. Geometric Sequence. A geometric sequence is a sequence that satisfies a n = a n-1 r where r is the common ratio. The common ratio r = or . Example 1:.
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Geometric Sequences and Series To recognize, write, and use geometric sequences
Geometric Sequence • A geometric sequence is a sequence that satisfies an = an-1r where r is the common ratio. • The common ratio r = or .
Example 1: • Determine whether or not the following sequence is geometric. If it is, find the common ratio. 60, 30, 0, 30, 60, . . . There is not a constant ratio, so it isn’t a geometric sequence.
Example 1b: • Find the common ratio of the following geometric sequences. 3, 6, 12, 24, … r = 2 1, ½, ¼, ⅛, … r = ½
The nth Term a1 = a1 a2 = a1r a3 = a2r = a1r·r = a1r2 a4 = a3r = a1r3 a5 = a4r= a1r4 an = a1rn-1
Geometric Sequences • Every geometric sequence can be written in the form a1, a1r, a1r2, a1r3, … a1rn-1 • A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers.
Example 2: • Write the first five terms of the geometric sequence whose first term is a1 = 5 and whose common ratio is - 3. 5, -15, 45, -135, 405
Example 3: • Find the eighteenth term of the geometric sequence that begins with 15, 12, 9.6, 7.68 . . . . 1 1 a18 = .3378