Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 12 Sequences, Series, and the Binomial Theorem

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 12.4 Partial Sums of Arithmetic and Geometric Sequences

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Partial Sum S n of an Arithmetic Sequence The partial sum S n of the first n terms of an arithmetic sequence is given by where a 1 is the first term of the sequence and a n is the nth term.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Use the partial sum formula to find the sum of the first four terms of the arithmetic sequence 3, 9, 15, 21, 27, … Example Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the sum of the first 25 even integers. Because 2, 4, 6, …, 50 is an arithmetic sequence, use the formula for S n is used with n = 25, a 1 = 2, and a n = 50. Example Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Partial Sum S n of a Geometric Sequence The partial sum S n of the first n terms of a geometric sequence is given by where a 1 is the first term of the sequence, r is the common ratio, and r  1.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the sum of the first five terms of the geometric sequence 3, 12, 48, 192, 768, 3072, … Example Solution Here n = 5, the first term a 1 = 3 and r = 4.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Chelsea made $20,000 during the first year she was self- employed. She made an additional 15% more than the previous year in each subsequent year. a. How much did she make during her fifth year of business? b. What were her total earnings during the five years? a.a n = a 1 r n – 1 a 5 = 20,000(1.15) 4  34, Chelsea’s earnings are modeled by a geometric sequence where n = 5, a 1 = 20,000, and r = 1.15 continued Example Solution

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall b. Chelsea made approximately $34, during her fifth year of self-employment, and a total of $134, during the first five years.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Sum of the Terms of an Infinite Geometric Sequence The sum S ∞ of the terms of an infinite geometric sequence is given by where a 1 is the first term of the sequence, r is the common ratio, and |r| < 1. If |r|  1, S ∞ does not exist.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Find the sum of the terms of the geometric sequence Example Solution For the sequence, r = 2/3.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall On its first pass, a pendulum swings through an arc whose length is 20 inches. On each pass thereafter, the arc length is 60% of the arc length on the preceding pass. Find the total distance the pendulum travels before it comes to rest. We must find the sum of the terms of the infinite geometric sequence whose first term a 1 = is 20 and whose common ratio, r, is 0.6. Example Solution The pendulum travels a total distance of 50 inches before it comes to a rest.