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Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759
BuyFind

Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759

Solutions

Chapter
Section
Chapter 12.6, Problem 59E
Textbook Problem
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DISCOVER ■ PROVE: Alternating Sums of Binomial Coefficients Find the sum

( n 0 ) ( n 1 ) + ( n 2 ) + ( 1 ) n ( n n )

by finding a pattern as in Exercise 58. Prove your result by expanding (1 − 1)n using the Binomial Theorem.

Expert Solution
To determine

The sum of the series (n0)(n1)+(n2)+(1)n(nn) .

Explanation of Solution

Given:

The series is (n0)(n1)+(n2)+(1)n(nn) .

Calculation:

Rewrite the series to convert the series in the form of Binomial expansion of (a+b)n .

(n0)(n1)+(n2)+(1)n(nn)=(n0)(1)n(n1)(1)n1(1)2+(nn)(1)n=[1+(1)]n=(11)n

Expand the term (11)n by Binomial Theorem

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Chapter 12 Solutions

Precalculus: Mathematics for Calculus (Standalone Book)
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Ch. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - Partial Sums Find the first six partial sums S1,...Ch. 12.1 - Partial Sums Find the first six partial sums S1,...Ch. 12.1 - Partial Sums Find the first six partial sums S1,...Ch. 12.1 - Partial Sums Find the first six partial sums S1,...Ch. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - Evaluating a Sum Find the sum. 47. k=14kCh. 12.1 - Evaluating a Sum Find the sum. 48. k=14k2Ch. 12.1 - Evaluating a Sum Find the sum. 49. k=131kCh. 12.1 - Evaluating a Sum Find the sum. 50. j=1100(1)jCh. 12.1 - Evaluating a Sum Find the sum. 51. i=18[1+(1)i]Ch. 12.1 - Evaluating a Sum Find the sum. 52. i=41210Ch. 12.1 - Evaluating a Sum Find the sum. 53. k=152k1Ch. 12.1 - Evaluating a Sum Find the sum. 54. i=13i2iCh. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - nth Term of a Sequence Find a formula for the nth...Ch. 12.1 - Compound Interest Julio deposits 2000 in a savings...Ch. 12.1 - Compound Interest Helen deposits 100 at the end of...Ch. 12.1 - Population of a City A city was incorporated in...Ch. 12.1 - Paying off a Debt Margarita borrows 10,000 from...Ch. 12.1 - Fish Farming A fish farmer has 5000 catfish in his...Ch. 12.1 - Price of a House The median price of a house in...Ch. 12.1 - Salary Increases A newly hired salesman is...Ch. 12.1 - Concentration of a Solution A biologist is trying...Ch. 12.1 - Fibonaccis Rabbits Fibonacci posed the following...Ch. 12.1 - DISCOVER PROVE: Different Sequences That Start the...Ch. 12.1 - DISCUSS: A Recursively Defined Sequence Find the...Ch. 12.1 - DISCUSS: A Different Type of Recursion Find the...Ch. 12.2 - An arithmetic sequence is a sequence in which the...Ch. 12.2 - The sequence given by an = a + (n 1)d is an...Ch. 12.2 - True or False? If False, give a reason. 3. The nth...Ch. 12.2 - True or False? If False, give a reason. 4. If we...Ch. 12.2 - Terms of an Arithmetic Sequence The nth term of an...Ch. 12.2 - Terms of an Arithmetic Sequence The nth term of an...Ch. 12.2 - Terms of an Arithmetic Sequence The nth term of an...Ch. 12.2 - Terms of an Arithmetic Sequence The nth term of an...Ch. 12.2 - Terms of an Arithmetic Sequence The nth term of an...Ch. 12.2 - Terms of an Arithmetic Sequence The nth term of an...Ch. 12.2 - nth Term of an Arithmetic Sequence Find the nth...Ch. 12.2 - nth Term of an Arithmetic Sequence Find the nth...Ch. 12.2 - nth Term of an Arithmetic Sequence Find the nth...Ch. 12.2 - nth Term of an Arithmetic Sequence Find the nth...Ch. 12.2 - nth Term of an Arithmetic Sequence Find the nth...Ch. 12.2 - nth Term of an Arithmetic Sequence Find the nth...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? The first four terms of a...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Partial Sums of an Arithmetic Sequence A partial...Ch. 12.2 - Adding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Adding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Special Triangle Show that a right triangle whose...Ch. 12.2 - Product of Numbers Find the product of the numbers...Ch. 12.2 - Harmonic Sequence A sequence is harmonic if the...Ch. 12.2 - Harmonic Mean The harmonic mean of two numbers is...Ch. 12.2 - Depreciation The purchase value of an office...Ch. 12.2 - Poles in a Pile Telephone poles are being stored...Ch. 12.2 - Salary Increases A man gets a job with a salary of...Ch. 12.2 - Drive-In Theater A drive-in theater has spaces for...Ch. 12.2 - Theater Seating An architect designs a theater...Ch. 12.2 - Falling Ball When an object is allowed to fall...Ch. 12.2 - The Twelve Days of Christmas In the well-known...Ch. 12.2 - DISCUSS: Arithmetic Means The arithmetic mean (or...Ch. 12.3 - A geometric sequence is a sequence in which the...Ch. 12.3 - The sequence given by an = arn1 is a geometric...Ch. 12.3 - True or False? If we know the first and second...Ch. 12.3 - (a) The nth partial sum of a geometric sequence an...Ch. 12.3 - nth Term of a Geometric Sequence The nth term of a...Ch. 12.3 - nth Term of a Geometric Sequence The nth term of a...Ch. 12.3 - nth Term of a Geometric Sequence The nth term of a...Ch. 12.3 - nth Term of a Geometric Sequence The nth term of a...Ch. 12.3 - nth Term of a Geometric Sequence Find the nth term...Ch. 12.3 - nth Term of a Geometric Sequence Find the nth term...Ch. 12.3 - nth Term of a Geometric Sequence Find the nth term...Ch. 12.3 - nth Term of a Geometric Sequence Find the nth term...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Geometric Sequence? Find the first five terms of...Ch. 12.3 - Geometric Sequence? Find the first five terms of...Ch. 12.3 - Geometric Sequence? Find the first five terms of...Ch. 12.3 - Geometric Sequence? Find the first five terms of...Ch. 12.3 - Geometric Sequence? Find the first five terms of...Ch. 12.3 - Geometric Sequence? Find the first five terms of...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Which Term? The first term of a geometric sequence...Ch. 12.3 - Which Term? The second and fifth terms of a...Ch. 12.3 - Partial Sums of a Geometric Sequence Find the...Ch. 12.3 - Partial Sums of a Geometric Sequence Find the...Ch. 12.3 - Partial Sums of a Geometric Sequence Find the...Ch. 12.3 - Partial Sums of a Geometric Sequence Find the...Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Partial Sums of a Geometric Sequence Find the sum....Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Repeated Decimal Express the repeating decimal as...Ch. 12.3 - Repeated Decimal Express the repeating decimal as...Ch. 12.3 - Repeated Decimal Express the repeating decimal as...Ch. 12.3 - Repeated Decimal Express the repeating decimal as...Ch. 12.3 - Repeated Decimal Express the repeating decimal as...Ch. 12.3 - Repeated Decimal Express the repeating decimal as...Ch. 12.3 - Geometric Means If the numbers a1, a2, , an form a...Ch. 12.3 - Partial Sum of a Geometric Sequence Find the sum...Ch. 12.3 - Arithmetic or Geometric? The first four terms of a...Ch. 12.3 - Arithmetic or Geometric? The first four terms of a...Ch. 12.3 - Depreciation A construction company purchases a...Ch. 12.3 - Family Tree A person has two parents, four...Ch. 12.3 - Bouncing Ball A ball is dropped from a height of...Ch. 12.3 - Bacteria Culture A culture initially has 5000...Ch. 12.3 - Mixing Coolant A truck radiator holds 5 gal and is...Ch. 12.3 - Musical Frequencies The frequencies of musical...Ch. 12.3 - Bouncing Ball A ball is dropped from a height of 9...Ch. 12.3 - Geometric Savings Plan A very patient woman wishes...Ch. 12.3 - St. Ives The following is a well-known childrens...Ch. 12.3 - Drug Concentration A certain drug is administered...Ch. 12.3 - Bouncing Ball A certain ball rebounds to half the...Ch. 12.3 - Bouncing Ball If the ball in Exercise 97 is...Ch. 12.3 - Geometry A circular disk of radius R is cut out of...Ch. 12.3 - PROVE: Reciprocals of a Geometric Sequence If a1,...Ch. 12.3 - PROVE: Logarithms of a Geometric Sequence If a1,...Ch. 12.3 - PROVE: Exponentials of an Arithmetic Sequence If...Ch. 12.3 - Geometry A yellow square of side 1 is divided into...Ch. 12.4 - An annuity is a sum of money that is paid in...Ch. 12.4 - The __________ __________ of an annuity is the...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Saving How much money should be invested every...Ch. 12.4 - Saving How much money should be invested monthly...Ch. 12.4 - Annuity What is the present value of an annuity...Ch. 12.4 - Annuity What is the present value of an annuity...Ch. 12.4 - Funding an Annuity How much money must be invested...Ch. 12.4 - Funding an Annuity A 55-year-old man deposits...Ch. 12.4 - Financing a Car A woman wants to borrow 12,000 to...Ch. 12.4 - Mortgage What is the monthly payment on a 30-year...Ch. 12.4 - Mortgage What is the monthly payment on a 30-year...Ch. 12.4 - Mortgage What is the monthly payment on a 15-year...Ch. 12.4 - Mortgage Dr. Gupta is considering a 30-year...Ch. 12.4 - Mortgage A couple can afford to make a monthly...Ch. 12.4 - Financing a Car Jane agrees to buy a car for a...Ch. 12.4 - Financing a Ring Mike buys a ring for his fiancee...Ch. 12.4 - Mortgage A couple secures a 30-year loan of...Ch. 12.4 - Mortgage A couple needs a mortgage of 300,000....Ch. 12.4 - Interest Rate John buys a stereo system for 640....Ch. 12.4 - Interest Rate Janets payments on her 12,500 car...Ch. 12.4 - Interest Rate An item at a department store is...Ch. 12.4 - Interest Rate A man purchases a 2000 diamond ring...Ch. 12.4 - DISCOVER: Present Value of an Annuity (a) Draw a...Ch. 12.4 - DISCOVER: An Annuity That Lasts Forever An annuity...Ch. 12.4 - DISCOVER: Amortizing a Mortgage When they bought...Ch. 12.5 - Mathematical induction is a method of proving that...Ch. 12.5 - Which of the following is true about Step 2 in a...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Formula for a Recursive Sequence A sequence is...Ch. 12.5 - Formula for a Recursive Sequence A sequence is...Ch. 12.5 - Proving a Factorization Show that x y is a factor...Ch. 12.5 - Proving a Factorization Show that x + y is a...Ch. 12.5 - Fibonacci Sequence Fn denotes the nth term of the...Ch. 12.5 - Fibonacci Sequence Fn denotes the nth term of the...Ch. 12.5 - Fibonacci Sequence Fn denotes the nth term of the...Ch. 12.5 - Fibonacci Sequence Fn denotes the nth term of the...Ch. 12.5 - Fibonacci Sequence Fn denotes the nth term of the...Ch. 12.5 - Formula Using Fibonacci Numbers Let an be the nth...Ch. 12.5 - Discover and Prove an Inequality Let Fn be the nth...Ch. 12.5 - Discover and Prove an Inequality Find and prove an...Ch. 12.5 - DISCUSS: True or False? Determine whether each...Ch. 12.5 - DISCUSS: All Cats Are Black? What is wrong with...Ch. 12.6 - An algebraic expression of the form a + b, which...Ch. 12.6 - We can find the coefficients in the expansion of...Ch. 12.6 - The binomial coefficients can be calculated...Ch. 12.6 - To expand (a + b)n, we can use the __________...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Pascals Triangle Use Pascals triangle to expand...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Binomial Theorem Use the Binomial Theorem to...Ch. 12.6 - Binomial Theorem Use the Binomial Theorem to...Ch. 12.6 - Binomial Theorem Use the Binomial Theorem to...Ch. 12.6 - Binomial Theorem Use the Binomial Theorem to...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Factoring Factor using the Binomial Theorem. 43....Ch. 12.6 - Factoring Factor using the Binomial Theorem. 44....Ch. 12.6 - Factoring Factor using the Binomial Theorem. 45....Ch. 12.6 - Factoring Factor using the Binomial Theorem. 46....Ch. 12.6 - Simplifying a Difference Quotient Simplify using...Ch. 12.6 - Simplifying a Difference Quotient Simplify using...Ch. 12.6 - Proving a Statement Show that the given statement...Ch. 12.6 - Proving a Statement Show that the given statement...Ch. 12.6 - Proving a Statement Show that the given statement...Ch. 12.6 - Proving a Statement Show that the given statement...Ch. 12.6 - Proving an Identity In this exercise we prove the...Ch. 12.6 - Proof Using Induction Prove that (rn)is an integer...Ch. 12.6 - Difference in Volumes of Cubes The volume of a...Ch. 12.6 - Probability of Hitting a Target The probability...Ch. 12.6 - DISCUSS: Powers of Factorials Which is larger,...Ch. 12.6 - DISCOVER PROVE: Sums of Binomial Coefficients Add...Ch. 12.6 - DISCOVER PROVE: Alternating Sums of Binomial...Ch. 12 - (a) What is a sequence? What notation do we use to...Ch. 12 - (a) What is a recursively defined sequence? (b)...Ch. 12 - (a) What is meant by the partial sums of a...Ch. 12 - (a) What is an arithmetic sequence? Write a...Ch. 12 - (a) What is a geometric sequence? Write an...Ch. 12 - (a) What is an infinite geometric series? (b) What...Ch. 12 - (a) Write 13 + 23 + 33 + 43 + 53 using sigma...Ch. 12 - (a) What is an annuity? Write an expression for...Ch. 12 - (a) State the Principle of Mathematical Induction....Ch. 12 - (a) Write Pascals triangle. How are the entries in...Ch. 12 - (a) What does the symbol n! mean? Find 5!. (b)...Ch. 12 - (a) State the Binomial Theorem. (b) Use the...Ch. 12 - Terms of a Sequence Find the first four terms as...Ch. 12 - Terms of a Sequence Find the first four terms as...Ch. 12 - Terms of a Sequence Find the first four terms as...Ch. 12 - Terms of a Sequence Find the first four terms as...Ch. 12 - Terms of a Sequence Find the first four terms as...Ch. 12 - Terms of a Sequence Find the first four terms as...Ch. 12 - Recursive Sequences A sequence is defined...Ch. 12 - Recursive Sequences A sequence is defined...Ch. 12 - Recursive Sequences A sequence is defined...Ch. 12 - Recursive Sequences A sequence is defined...Ch. 12 - Arithmetic or Geometric? The nth term of a...Ch. 12 - Arithmetic or Geometric? The nth term of a...Ch. 12 - Arithmetic or Geometric? The nth term of a...Ch. 12 - Arithmetic or Geometric? The nth term of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Proving a Sequence Is Geometric Show that 3, 6i,...Ch. 12 - nth Term of a Geometric Sequence Find the nth term...Ch. 12 - Finding Terms of Arithmetic and Geometric...Ch. 12 - Finding Terms of Arithmetic and Geometric...Ch. 12 - Finding Terms of Arithmetic and Geometric...Ch. 12 - Finding Terms of Arithmetic and Geometric...Ch. 12 - Salary A teacher makes 32,000 in his first year at...Ch. 12 - Salary A colleague of the teacher in Exercise 29,...Ch. 12 - Bacteria Culture A certain type of bacteria...Ch. 12 - Arithmetic Sequences If a1, a2, a3, and b1, b2,...Ch. 12 - Geometric Sequences If a1, a2, a3, and b1, b2, b3,...Ch. 12 - Arithmetic or Geometric? (a) If a1, a2, a3, is an...Ch. 12 - Arithmetic and Geometric Sequences Find the values...Ch. 12 - Arithmetic and Geometric Sequences Find the values...Ch. 12 - Partial Sums Find the sum. 37. k=36(k+1)2Ch. 12 - Partial Sums Find the sum. 38. i=142i2i1Ch. 12 - Partial Sums Find the sum. 39.k=16(k+1)2k1Ch. 12 - Partial Sums Find the sum. 40. m=153m2Ch. 12 - Sigma Notation Write the sum without using sigma...Ch. 12 - Sigma Notation Write the sum without using sigma...Ch. 12 - Sigma Notation Write the sum without using sigma...Ch. 12 - Sigma Notation Write the sum without using sigma...Ch. 12 - Sigma Notation Write the sum using sigma notation....Ch. 12 - Sigma Notation Write the sum using sigma notation....Ch. 12 - Sigma Notation Write the sum using sigma notation....Ch. 12 - Sigma Notation Write the sum using sigma notation....Ch. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Terms of an Arithmetic Sequence The first term of...Ch. 12 - Terms of an Geometric Sequence The sum of the...Ch. 12 - Ancestors A person has two parents, four...Ch. 12 - Annuity Find the amount of an annuity consisting...Ch. 12 - Investment How much money should be invested every...Ch. 12 - Mortgage What are the monthly payments on a...Ch. 12 - Mathematical Induction Use mathematical induction...Ch. 12 - Mathematical Induction Use mathematical induction...Ch. 12 - Mathematical Induction Use mathematical induction...Ch. 12 - Proof by Induction Use mathematical induction to...Ch. 12 - Proof by Induction Use mathematical induction to...Ch. 12 - Proof by Induction Use mathematical induction to...Ch. 12 - Binomial Coefficients Evaluate the expression. 73....Ch. 12 - Binomial Coefficients Evaluate the expression....Ch. 12 - Binomial Coefficients Evaluate the expression. 75....Ch. 12 - Binomial Coefficients Evaluate the expression. 76....Ch. 12 - Binomial Expansion Expand the expression. 77. (A ...Ch. 12 - Binomial Expansion Expand the expression. 78. (x +...Ch. 12 - Binomial Expansion Expand the expression. 79. (1 ...Ch. 12 - Binomial Expansion Expand the expression. 80. (2x...Ch. 12 - Terms in a Binomial Expansion Find the indicated...Ch. 12 - Terms in a Binomial Expansion Find the indicated...Ch. 12 - Terms in a Binomial Expansion Find the indicated...Ch. 12 - Find the first six terms and the sixth partial sum...Ch. 12 - A sequence is defined recursively by an+1 = 3an ...Ch. 12 - An arithmetic sequence begins 2, 5, 8, 11, 14, ....Ch. 12 - A geometric sequence begins 12,3,34,316,364,. (a)...Ch. 12 - The first term of a geometric sequence is 25, and...Ch. 12 - The first term of an arithmetic sequence is 10,...Ch. 12 - Let a1, a2, a3, be a geometric sequence with...Ch. 12 - Write the expression without using sigma notation,...Ch. 12 - Find the sum. (a) 13+232+2233+2334++29310 (b)...Ch. 12 - Expand (2x + y2)5.Ch. 12 - Find the term containing x3 in the binomial...Ch. 12 - A puppy weighs 0.85 lb at birth, and each week he...Ch. 12 - Retirement Accounts Many college professors keep...Ch. 12 - Fitness Program Sheila decides to embark on a...Ch. 12 - Monthly Savings Program Alice opens a savings...Ch. 12 - Pollution A chemical plant discharges 2400 tons of...Ch. 12 - Annual Savings Program Ursula opens a 1-year CD...

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