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CalculusPrecalculus: Mathematics for Calculus (Standalone Book)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.BuyFind*arrow_forward*

7th Edition

James Stewart + 2 others

Publisher: Brooks Cole

ISBN: 9781305071759

Chapter 12.6, Problem 59E

Textbook Problem

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**DISCOVER ■ PROVE: Alternating Sums of Binomial Coefficients** Find the sum

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

**Given:**

The series is

**Calculation:**

Rewrite the series to convert the series in the form of Binomial expansion of

Expand the term

Precalculus: Mathematics for Calculus (Standalone Book)

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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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