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How do I find the sum & terms of geometric sequences and series? Essential Question: What is a sequence and how do I find its terms and sums? How do I find the sum & terms of geometric sequences and series?

Geometric Sequence Geometric Sequence– a sequence whose consecutive terms have a common ratio. You multiply to get the next term.

Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. The number r is the common ratio.

Formula for a Geometric Sequence an = a1r n – 1

Ex. 1 2, 4, 8, 16, …, formula?, … 12, 36, 108, 324, …, formula?, … Are these geometric? If so, find the formula. 2, 4, 8, 16, …, formula?, … an = 2(2)n – 1 12, 36, 108, 324, …, formula?, … an = 12(3)n – 1 an = (-1/3)(-1/3) n – 1 1, 4, 9, 16, …, formula? , … Not geometric, but what is the formula?

Ex. 2 Write the first five terms of the geometric sequence whose first term is a1 = 9 and r = (1/3). an = a1rn – 1

Ex. 3 Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05 an = a1rn – 1 a15 = (20)(1.05)15 – 1 a15 = 39.599

Ex. 4 Find a formula for the nth term. 5, 15, 45, … an = a1rn – 1 an = 5(3)n – 1 What is the 9th term? an = 5(3)n – 1 a9 = 5(3)8 a9 = 32805

Ex. 5 Find the first 3 terms an = 36(3)n – 3 a1 = 36(3)1-3 4, 12, 36

Ex. 6 The fourth term of a geometric sequence is 8, and the 7th term is 1. Find the geometric rule. an = a1rn-1 a4 = 8 a7 = a4r7-4 a7 = 1

Sum of a finite geometric series

Ex. 7 Find the sum: = 1.714

Find the value of n for the geometric series Ex. 8 Find the value of n for the geometric series Take the log base 6 of both sides or use the definition of a log to solve for n

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Homework: WS #1-39 odd