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Published byQuentin Wilkinson Modified over 8 years ago

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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

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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) = –559872.

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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(0.0000128) = 0.00064.

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Sum of the First n Terms of a Geometric Sequence

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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

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Example Solution Determine whether each series has a limit. If a limit exists, find it. a) 2 – 12 – 72 – 4323 – · · · b) 50 + 10 + 2 + 0.4 + 0.08 + · · · a) Here r = 6, so | r | = | 6 | = 6. Since | r | > 1, the series does not have a limit.

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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.

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Example Solution Find the fraction notation for 0.482482482…. We can express this as 0.482 + 0.000482 + 0.000000482 + · · ·. This is an infinite geometric series, where a 1 = 0.482 and r = 0.001. Since | r | < 1, this series has a limit: Thus fraction notation for 0.482482482… is

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

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