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: 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 a 1 = a 1 a 2 = a 1 r a 3 = a 2 r = a 1 r·r = a 1 r 2 a 4 = a 3 r = a 1 r 3 a 5 = a 4 r = a 1 r 4 a n = a 1 r n-1

Geometric Sequences Every geometric sequence can be written in the form a 1, a 1 r, a 1 r 2, a 1 r 3, … a 1 r n-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 a 1 = 5 and whose common ratio is , -15, 45, -135, 405

Example 3: Find the eighteenth term of the geometric sequence that begins with 15, 12, 9.6, a 18 =