Example Solution For each geometric sequence, find the common ratio. a)  2,  12,  72,  432,... b) 50, 10, 2, 0.4, 0.08,... SequenceCommon Ratio a)  2,  12,  72,  432,... b) 50, 10, 2, 0.4, 0.08,... r = 6 r = 0.2

Example Solution Find the 8 th term of each sequence. a) –2, –12, –72, –432, –2592,... b) 50, 10, 2, 0.4, 0.08,... a) First, we note that a 1 = –2, n = 8, and r = 6. The formula a n = a 1 r n – 1 gives us a 8 = –2·6 8 – 1 = –2·6 7 = –2(279936) = –

Solution continued b) First, we note that a 1 = 50, n = 8, and r = 0.2. The formula a n = a 1 r n – 1 gives us a 8 = 50·(0.2) 8 – 1 = 50·(0.2) 7 = 50( ) =

Sum of the First n Terms of a Geometric Sequence

Example Solution Find the sum of the first 9 terms of the geometric sequence  1, 4,  16, 64,.... First, we note that a 1 =  1,n = 9, and Then, substituting in the formula we have

Example Solution Determine whether each series has a limit. If a limit exists, find it. a)  2 – 12 – 72 – 4323 – · · · b) · · · a) Here r = 6, so | r | = | 6 | = 6. Since | r | > 1, the series does not have a limit.

Solution continued We find the limit by substituting into the formula for S ∞ : b) Here r = 0.2, so | r | = | 0.2 | = 0.2. Since | r | < 1, the series does have a limit.

Example Solution Find the fraction notation for …. We can express this as · · ·. This is an infinite geometric series, where a 1 = and r = Since | r | < 1, this series has a limit: Thus fraction notation for … is

Problem Solving For some problem-solving situations, the translation may involve geometric sequences or series.