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Copyright © 2014, 2010 Pearson Education, Inc. Chapter 9 Further Topics in Algebra Copyright © 2014, 2010 Pearson Education, Inc.

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Section 9.1 Sequences ands Series 1. Use sequence notation and find specific and general terms in a sequence. 2. Use factorial notation. 3. Use summation notation to write partial sums of a series. SECTION 1.1

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Copyright © 2014, 2010 Pearson Education, Inc. 3 DEFINITION OF A SEQUENCE An infinite sequence is a function whose domain is the set of positive integers. The function values, written as a 1, a 2, a 3, a 4, …, a n, …, are called the terms of the sequence. The nth term, a n, is called the general term of the sequence.

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Copyright © 2014, 2010 Pearson Education, Inc. 4 Write the first six terms of the sequence defined by: Replace n with each integer from 1 to 6. Example: Writing the Terms of a Sequence from the General Term

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Copyright © 2014, 2010 Pearson Education, Inc. 5 Example: Writing the Terms of a Sequence from the General Term

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Copyright © 2014, 2010 Pearson Education, Inc. 6 Write the general term a n for a sequence whose first five terms are given. Write the position number of the term above each term of the sequence and look for a pattern that connects the term to the position number of the term. Example: Finding a General Term of a Sequence from a Pattern

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Copyright © 2014, 2010 Pearson Education, Inc. 7 a. Apparent pattern: Here 1 = 1 2, 4 = 2 2, 9 = 3 2, 16 = 4 2, and 25 =5 2. Each term is the square of the position number of that term. This suggests a n = n 2. Example: Finding a General Term of a Sequence from a Pattern

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Copyright © 2014, 2010 Pearson Education, Inc. 8 b. Apparent pattern: When the terms alternate in sign and n = 1, we use factors such as (−1) n if we want to begin with the factor −1 or we use factors such as (−1) n+1 if we want to begin with the factor 1. Example: Finding a General Term of a Sequence from a Pattern

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Copyright © 2014, 2010 Pearson Education, Inc. 9 Notice that each term can be written as a quotient with denominator equal to the position number and numerator equal to one less than the position number, suggesting the general term Example: Finding a General Term of a Sequence from a Pattern

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Copyright © 2014, 2010 Pearson Education, Inc. 10 DEFINITION OF FACTORIAL For any positive integer n, n factorial (written n!) is defined as As a special case, zero factorial (written 0!) is defined as

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Copyright © 2014, 2010 Pearson Education, Inc. 11 Simplify. = 16 · 15 = 240 = (n + 1)n Example: Simplifying a Factorial Expression

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Copyright © 2014, 2010 Pearson Education, Inc. 12 Write the first five terms of the sequence whose general term is: Replace n with each integer from 1 through 5. Example: Writing Terms of a Sequence Involving Factorials

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Copyright © 2014, 2010 Pearson Education, Inc. 13 Example: Writing Terms of a Sequence Involving Factorials

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Copyright © 2014, 2010 Pearson Education, Inc. 14 SUMMATION NOTATION The sum of the first n terms of a sequence a 1, a 2, a 3, …, a n, … is denoted by The letter i in the summation notation is called the index of summation, n is called the upper limit, and 1 is called the lower limit, of the summation.

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Copyright © 2014, 2010 Pearson Education, Inc. 15 Find each sum. a.Replace i with integers 1 through 9, inclusive, and then add. Example: Evaluating Sums Given in Summation Notation

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Copyright © 2014, 2010 Pearson Education, Inc. 16 b.Replace j with integers 4 through 7, inclusive, and then add. Example: Evaluating Sums Given in Summation Notation

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Copyright © 2014, 2010 Pearson Education, Inc. 17 c.Replace k with integers 0 through 4, inclusive, and then add. Example: Evaluating Sums Given in Summation Notation

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Copyright © 2014, 2010 Pearson Education, Inc. 18 SUMMATION PROPERTIES Let a k and b k, represent the general terms of two sequences, and let c represent any real number. Then

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Copyright © 2014, 2010 Pearson Education, Inc. 19 SUMMATION PROPERTIES

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Copyright © 2014, 2010 Pearson Education, Inc. 20 DEFINITION OF A SERIES Let a 1, a 2, a 3, …, a k, … be an infinite sequence. Then 1. The sum of the first n terms of the sequence is called the nth partial sum of the sequence and is denoted by This sum is a finite series.

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Copyright © 2014, 2010 Pearson Education, Inc. 21 DEFINITION OF A SERIES 2.The sum of all terms of the infinite sequence is called an infinite series and is denoted by

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Copyright © 2014, 2010 Pearson Education, Inc. 22 Write each sum in summation notation. a.This is the sum of consecutive odd integers from 3 to 21. Each can be expressed as 2k + 1, starting with k = 1 and ending with k = 10. Example: Writing a Partial Sum in Summation Notation

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Copyright © 2014, 2010 Pearson Education, Inc. 23 b.This finite series is the sum of fractions, each of which has numerator 1 and denominator k 2, starting with k = 2 and ending with k = 7. Example: Writing a Partial Sum in Summation Notation

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Copyright © 2014, 2010 Pearson Education, Inc. 24 Section 9.5 The Binomial Theorem 1. Use Pascal’s Triangle to compute binomial coefficients. 2. Use Pascal’s Triangle to expand a binomial power. 3. Use the Binomial Theorem to expand a binomial power. 4. Find the coefficient of a term in a binomial expansion. SECTION 1.1

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Copyright © 2014, 2010 Pearson Education, Inc. 25 BINOMIAL EXPANSIONS The patterns for expansions of (x+ y) n (with n = 1, 2, 3, 4, 5) suggest the following: 1. The expansion of (x+ y) n has n + 1 terms. 2. The sum of the exponents on x and y in each term equals n. 3. The exponent on x starts at n (x n = x n · y 0 ) in the first term and decreases by 1 for each term until it is 0 in the last term (x 0 · y n = y n ).

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Copyright © 2014, 2010 Pearson Education, Inc. 26 BINOMIAL EXPANSIONS 4. The exponent on y starts at 0 in the first term (x n = x n · y 0 ) and increases by 1 for each term until it is n in the last term (x 0 · y n = y n ). 5. The variables x and y have symmetrical roles. That is, replacing x with y and y with x in the expansion of (x+ y) n yields the same terms, just in a different order.

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Copyright © 2014, 2010 Pearson Education, Inc. 27 BINOMIAL EXPANSIONS You may also have noticed that the coefficients of the first and last terms are both 1 and the coefficients of the second and the next-to-last terms are equal. In general, the coefficients of x n−j y j and x j y n−j are equal for j = 0, 1, 2, …, n. The coefficients in a binomial expansion of (x + y) n are called the binomial coefficients.

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Copyright © 2014, 2010 Pearson Education, Inc. 28 PASCAL ’ S TRIANGLE When expanding (x + y) n the coefficients of each term can be determined using Pascal’s Triangle. The top row of the triangle, which contains only the number 1, represents the coefficients of (x + y) 0 and is referred to as the zeroth row. The next row, called the first row, represents the coefficients of (x + y) 1. Each row begins and ends with 1. Each entry of Pascal’s Triangle is found by adding the two neighboring entries in the previous row.

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Copyright © 2014, 2010 Pearson Education, Inc. 29 PASCAL ’ S TRIANGLE

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Copyright © 2014, 2010 Pearson Education, Inc. 30 Expand (4y – 2x) 5. The fifth row of Pascal’s Triangle yields the binomial coefficients 1, 5, 10, 10, 5, 1. Replace x with 4y and y with –2x. Example: Using Pascal’s Triangle to Expand a Binomial Power

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Copyright © 2014, 2010 Pearson Education, Inc. 31 Expanding a difference results in alternating signs. Example: Using Pascal’s Triangle to Expand a Binomial Power

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Copyright © 2014, 2010 Pearson Education, Inc. 32 DEFINITION OF If r and n are integers with 0 ≤ r ≤ n, then we define

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Copyright © 2014, 2010 Pearson Education, Inc. 33 Evaluate each binomial coefficient. Example: Evaluating

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Copyright © 2014, 2010 Pearson Education, Inc. 34 Example: Evaluating

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Copyright © 2014, 2010 Pearson Education, Inc. 35 THE BINOMIAL THEOREM If n is a natural number, then the binomial expansion of (x + y) n is given by The coefficient of x n–r y r is

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Copyright © 2014, 2010 Pearson Education, Inc. 36 Find the binomial expansion of (x – 3y) 4. (x – 3y) 4 = [x + (–3y)] 4 Example: Expanding a Binomial Power by Using the Binomial Theorem

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Copyright © 2014, 2010 Pearson Education, Inc. 37 Example: Expanding a Binomial Power by Using the Binomial Theorem

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Copyright © 2014, 2010 Pearson Education, Inc. 38 Example: Expanding a Binomial Power by Using the Binomial Theorem

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Copyright © 2014, 2010 Pearson Education, Inc. 39 PARTICULAR TERM IN A BINOMIAL EXPRESSION The term containing the factor x r in the expansion of (x + y) n is

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Copyright © 2014, 2010 Pearson Education, Inc. 40 Find the term containing x 10 in the expansion of (x + 2a) 15. Use the formula for the term containing x r. Example: Finding a Particular Term in a Binomial Expansion

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Copyright © 2014, 2010 Pearson Education, Inc. 41 Example: Finding a Particular Term in a Binomial Expansion

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