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Geometric Sequences & Series MATH 109 - Precalculus S. Rook Overview • Section 9.3 in the textbook: – Geometric sequences – Partial sums of finite geometric sequences – Infinite geometric series 2 Geometric Sequences Geometric Sequences • Geometric sequence: a sequence where the ratio of ANY two successive terms is equal to the same constant value for all natural numbers i where r is known as the common ratio – e.g.:1,2,4,8,,2n1 a1 = 1 and r = 2 – e.g.: 4,2,1, 1 ,,4 1 a1 = 4 and r = ½ r ai 1 ai n 1 2 2 4 Geometric Sequences (Continued) • The formula for the nth term of a geometric sequence is an a1r n1 where a1 is the first term of the sequence and r is the common ratio 5 Geometric Sequences (Example) Ex 1: Find the indicated term using the given geometric sequence: a) 80, 20, 5, … ; 6th b) -3, 9, -27, … ; 11th 6 Partial Sums of Finite Geometric Sequences Partial Sums of Finite Geometric Sequences • The nth partial sum of a geometric sequence is 1 r n given by where a is the first 1 , r 1 S n a1 term and r is the 1 r common ratio – Do not need to worry about deriving the formula – Just know how to use it • Also known as a Finite Geometric Series 8 Partial Sums of Finite Geometric Sequences (Example) Ex 2: Evaluate the partial sum for the given finite geometric sequence: a) n = 8 where the nth term is given by an = 7(2)n-1 b) n = 12 where the sixth term is 4⁄125 and the seventh term is 4⁄625 9 Infinite Geometric Series Infinite Geometric Series • Infinite Geometric Series: a summation of ALL the terms of an infinite geometric sequence • If |r| < 1, you will see in Calculus that rn will approach zero (rn → 0) as n approaches positive infinity (n → +oo) • Thus, the sum of an infinite geometric series is a1 S a1r 1 r i 1 i – Again, just know how to use the formula to solve problems 11 Infinite Geometric Series (Example) Ex 3: Evaluate the infinite geometric series: 1 a) 4 n 0 4 n b) 3 0.9 n n 0 12 Infinite Geometric Series (Example) Ex 4: Find the rational number representation of 0.36 13 Summary • After studying these slides, you should be able to: – Identify whether a given series is geometric – Evaluate the partial sum of a geometric sequence – Evaluate an infinite geometric series • Additional Practice – See the list of suggested problems for 9.3 • Next lesson – Study for the Final Exam! 14