Sequences And Series Cumulative Assessment
Question 1.The frequencies of the notes on a piano form a geometric sequence. The frequencies of G and A are shown in the diagram. What is the approximate frequency of E at ?Answer:
Question 2.You take out a loan for $16,000 with an interest rate of 0.75% per month. At the end of each month, you make a payment of $300.a. Write a recursive rule for the balance an of the loan at the beginning of the nth month.b. How much do you owe at the beginning of the 18th month?c. How long will it take to pay off the loan?d. If you pay $350 instead of $300 each month, how long will it take to pay off the loan? How much money will you save? Explain.Answer:
Question 3.The table shows that the force F needed to loosen a certain bolt with a wrench depends on the length of the wrenchs handle. Write an equation that relates and F. Describe the relationship.Answer:
Question 4.Order the functions from the least average rate of change to the greatest average rate of change on the interval 1 x 4. Justify your answers.Answer:
Question 6.The diagram shows the bounce heights of a basketball and a baseball dropped from a height of 10 feet. On each bounce, the basketball bounces to 36% of its previous height, and the baseball bounces to 30% of its previous height. About how much greater is the total distance traveled by the basketball than the total distance traveled by the baseball?A. 1.34 feet
Find An Arithmetic Series With 8 Terms And A Sum Of 324
Write an arithmetic series with 8 terms and a sum of 324.
Concept Used:
We have to determine an arithmetic sequence a
The formula for the sum of any arithmetic series is S
We have to determine an arithmetic sequence a
The formula for the sum of any arithmetic series is S
an arbitrary value and find the common difference d:
:16+23+30+37+44+51+58+65=324
Thus,
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Glencoe Algebra 2 Student Edition C2014
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PREALGEBRA
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Sum Of An Arithmetic Series
The sum Sn of the first n terms of an arithmetic series is given by the formula n n Sn or Sn
Example
The sum is an arithmetic series with common difference 3. Substituting k 1 and k 18 into the expression 3k 4 gives a1 3 4 7 and a18 3 4 58. There are 18 terms in the series, so n 18. Use the formula for the sum of an arithmetic series. Sn Sum formula n 18, a1 7, an 58 Simplify. Multiply.
Example 1 Find S for the n arithmetic series with a1 14, an 101, and n 30. Use the sum formula for an arithmetic series.Formula for nth term
Example 2 Find the sum of all positive odd integers less than 180. The series is 1 3 5 179. Find n using the formula for the nth term of an arithmetic sequence.n 2 18 S18 2
Sum formula
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Finding Sums Of Infinite Geometric Series 84 Exercises
Vocabulary and Core Concept CheckQuestion 1.The sum Sn of the first n terms of an infinite series is called a ________.Answer:
Question 2.WRITINGExplain how to tell whether the series \a1ri1 has a sum.Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 36, consider the infinite geometric series. Find and graph the partial sums Sn for n= 1, 2, 3, 4, and 5. Then describe what happens to Sn as n increases.Question 3.\Answer:
Question 4.\Answer:
4 + \Answer:
2 + \Answer:
In Exercises 714, find the sum of the infinite geometric series, if it exists.Question 7.\^\)Answer:
\^\)Answer:
\^\)Answer:
\^\)Answer:
2 + \Answer:
-5 2 \Answer:
3 + \Answer:
\Answer:
ERROR ANALYSIS In Exercises 15 and 16, describe and correct the error in finding the sum of the infinite geometric series.Question 15.
Question 17.MODELING WITH MATHEMATICSYou push your younger cousin on a tire swing one time and then allow your cousin to swing freely. On the first swing, your cousin travels a distance of 14 feet. On each successive swing, your cousin travels 75% of the distance of the previous swing. What is the total distance your cousin swings?Answer:
Question 18.MODELING WITH MATHEMATICSA company had a profit of $350,000 in its first year. Since then, the companys profit has decreased by 12% per year. Assuming this trend continues, what is the total profit the company can make over the course of its lifetime? Justify your answer.Answer:
Answer:
Sequences And Series Chapter Review
8.1 Defining and Using Sequences and Series
Question 1.Describe the pattern shown in the figure. Then write a rule for the nth layer of the figure, where n = 1 represents the top layer.Answer:
Write the series using summation notation.Question 2.
Question 12.Find the sum [la
You take a job with a starting salary of $37,000. Your employer offers you an annual raise of $1500 for the next 6 years. Write a rule for your salary in the nth year. What are your total earnings in 6 years?Answer:
8.3 Analyzing Geometric Sequences and Series
Question 14.Tell whether the sequence 7, 14, 28, 56, 112, . . . is geometric. Explain your reasoning.Answer:
Write a rule for the nth term of the geometric sequence. Then graph the first six terms of the sequence.Question 15.25, 10, 4, \ , . . .Answer:
Find the sum \5i1 .Answer:
8.4 Finding Sums of Infinite Geometric Series
Question 19.Consider the infinite geometric series 1, \ Find and graph the partial sums Sn for n= 1, 2, 3, 4, and 5. Then describe what happens to Sn as n increases.Answer:
Question 20.Find the sum of the infinite geometric series 2 + \, if it exists.Answer:
Write the repeating decimal 0.1212 . . . as a fraction in simplest form.Answer:
8.5 Using Recursive Rules with Sequences
Write the first six terms of the sequence.Question 22.a1 = 7, an = an-1 + 11Answer:
a1 = 26, an = \an-1.Answer:
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Unit : Sequences & Series
Flipped ClassroomIn your final unit of trigonometry, you will learn the material in a slightly different manner, on your own and at home. One day, youll come across a class where perhaps you have to do this merely because you do not understand your professor or perhaps you prefer not to make the trek to campus in the frigid arctic temperatures. To prepare you for this, you will complete the final unit of trigonometry by learning at home and practicing at school in front of your teacher. How will you be graded during this unit? Well, of course you will have a test at the end of the unit but to ensure that you are taking notes and completing the requirements of your at home classroom, you will be graded when you arrive to class each day based upon the following: ! Quality of the notes from the Lesson Videos ! Quality of the guided note pages ! Quality of any other requirements of the at home lesson ! Quality and Quantity of Comments/Questions/Answers made in Unit 6 Discussion Forum ! Effort put forth to Comprehend/Learn/Understand the material **The Flipped Classroom Unit will count as your project for the unit**
Total: 50 points
Learning Target: I can 1. Find the nth term of an arithmetic or geometric sequence 2. Find the position of a given term of an arithmetic or geometric sequence 3. Find sums of a finite arithmetic or geometric series 4. Use sequences and series to solve real-world problems 5. Use sigma notation to express sums
Resources:! ! ! !
Sequences And Series Chapter Test
Find the sum.\Answer:
\n2Answer:
\2k1Answer:
\4i1Answer:
Determine whether the graph represents an arithmetic sequence, geometric sequence, or neither. Explain your reasoning. Then write a rule for the nth term.Question 5.
2, 0, 3, 7, 12, . . .Answer:
Question 11.Write a recursive rule for the sequence 5, 20, 80, 320, 1280, . . .. Then write an explicit rule for the sequence using your recursive rule.Answer:
Question 12.The numbers a, b, and c are the first three terms of an arithmetic sequence. Is b half of the sum of a and c? Explain your reasoning.Answer:
Use the pattern of checkerboard quilts shown.a. What does n represent for each quilt? What does an represent?b. Make a table that shows n and an for n= 1, 2, 3, 4, 5, 6, 7, and 8.c. Use the rule an = \ to find an for n = 1, 2, 3, 4, 5, 6, 7, and 8.Compare these values to those in your table in part . What can you conclude? Explain.Answer:
Question 14.During a baseball season, a company pledges to donate $5000 to a charity plus $100 for each home run hit by the local team. Does this situation represent a sequence or a series? Explain your reasoning.Answer:
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The Sum Of The Series Is 145733
ExercisesFind Sn for each geometric series described. 1. a1 2, an 486, r 3 2. a1 1200, an 75, r 1 2
3. a1 , an 125, r 5
1 25
4. a1 3, r , n 41 3
5. a1 2, r 6, n 4
156.246. a1 2, r 4, n 6
4.447. a1 100, r , n 51 2
8. a3 20, a6 160, n 8
27309. a4 16, a7 1024, n 10
68.75
12 63a1 441 1 441 a1 63
a 1r a1
Lesson 82 Analyzing Arithmetic Sequences And Series
Essential Question How can you recognize an arithmetic sequence from its graph?In an arithmetic sequence, the difference of consecutive terms, called the common difference, is constant. For example, in the arithmetic sequence 1, 4, 7, 10, . . . , the common difference is 3.
EXPLORATION 1
Recognizing Graphs of Arithmetic SequencesWork with a partner. Determine whether each graph shows an arithmetic sequence. If it does, then write a rule for the nth term of the sequence, and use a spreadsheet to fond the sum of the first 20 terms. What do you notice about the graph of an arithmetic sequence?
EXPLORATION 2
Finding the Sum of an Arithmetic SequenceWork with a partner. A teacher of German mathematician Carl Friedrich Gauss asked him to find the sum of all the whole numbers from 1 through 100. To the astonishment of his teacher, Gauss came up with the answer after only a few moments. Here is what Gauss did:Explain Gausss thought process. Then write a formula for the sum Sn of the first n terms of an arithmetic sequence. Verify your formula by finding the sums of the first 20 terms of the arithmetic sequences in Exploration 1. Compare your answers to those you obtained using a spreadsheet.
Communicate Your Answer
How can you recognize an arithmetic sequence from its graph?Answer:
Find the sum of the terms of each arithmetic sequence.a. 1, 4, 7, 10, . . . , 301b. 1, 2, 3, 4, . . . , 1000c. 2, 4, 6, 8, . . . , 800Answer:
Which is different? Find both answers.Answer:
\Answer:
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The First Term Of The Series Is 7
976 a1 972 a1 4
The first term of the series is 4. Find a4 in a geometric series for which Sn 796.875, r , and n 8. 2 First use the sum formula to find a1.1 Sn
a 1r
Sum formula
a1 1 2 796.875 1 1 2
S8 796.875, r , n 8
Geometric SeriesFind Sn for each geometric series described. 1. a1 2, a6 64, r 2 126 3. a1 3, an 192, r 2 129 5. a1 3, an 3072, r 4 2457 7. a1 5, r 3, n 9 49,205 9. a1 6, r 3, n 7 3282 11. a1 , r 3, n 10 Find the sum of each geometric series. 13. 162 54 18 to 6 terms 15. 64 96 144 to 7 terms 463 17. n 1 1640n1 6 8 9
2. a1 160, a6 5, r 315 4. a1 81, an 16, r 55 6. a1 54, a6 , r 8. a1 6, r 1, n 21 6 10. a1 9, r , n 4 12. a1 16, r 1.5, n 6 66.52 3 2 9 1 728 3 2 3
Find the indicated term for each geometric series described. 23. Sn 1023, an 768, r 4 a1 3 25. Sn 1365, n 12, r 2 a1 1 24. Sn 10,160, an 5120, r 2 a1 80 26. Sn 665, n 6, r 1.5 a1 32
27. CONSTRUCTION A pile driver drives a post 27 inches into the ground on its first hit. Each additional hit drives the post the distance of the prior hit. Find the total distance the post has been driven after 5 hits.2 3
70 in.
28. COMMUNICATIONS Hugh Moore e-mails a joke to 5 friends on Sunday morning. Each of these friends e-mails the joke to 5 of her or his friends on Monday morning, and so on. Assuming no duplication, how many people will have heard the joke by the end of Saturday, not including Hugh? 97,655 people
Glencoe/McGraw-Hill
Arithmetic Means
Arithmetic Sequences
Sequences And Series 81 83 Quiz
Describe the pattern, write the next term, and write a rule for the nth term of the sequence.Question 1.
\Answer:
\11k2Answer:
\4 )i+3Answer:
Question 18.Pieces of chalk are stacked in a pile. Part of the pile is shown. The bottom row has 15 pieces of chalk, and the top row has 6 pieces of chalk. Each row has one less piece of chalk than the row below it. How many pieces of chalk are in the pile?Answer:
Question 19.You accept a job as an environmental engineer that pays a salary of $45,000 in the first year. After the first year, your salary increases by 3.5% per year.a. Write a rule giving your salary an for your nth year of employment.b. What will your salary be during your fifth year of employment?c. You work 10 years for the company. What are your total earnings? Justify your answer.Answer:
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Big Ideas Math Book Algebra 2 Answer Key Chapter 8 Sequences And Series
Check out Big Ideas Math Algebra 2 Answers Chapter 8 Sequences and Series aligned as per the Big Ideas Math Textbooks. Use the below available links for learning the Topics of BIM Algebra 2 Chapter 8 Sequences and Series easily and quickly. You just need to tap on them and avail the underlying concepts in it and score better grades in your exams. The Solutions covered here include Questions from Chapter Tests, Review Tests, Cumulative Practice, Cumulative Assessments, Exercise Questions, etc.
\x 8 = 17Answer:8)x = \Answer:Question 10.ABSTRACT REASONINGThe graph of the exponential decay function f = bx has an asymptote y = 0. How is the graph of f different from a scatter plot consisting of the points , , , . . .? How is the graph of f similar?Answer:
N 10 An 41 Sn 230 5 9 13
28. BUSINESS A merchant places $1 in a jackpot on August 1, then draws the name of a regular customer. If the customer is present, he or she wins the $1 in the jackpot. If the customer is not present, the merchant adds $2 to the jackpot on August 2 and draws another name. Each day the merchant adds an amount equal to the day of the month. If the first person to win the jackpot wins $496, on what day of the month was her or his name drawn? August 31Glencoe Algebra 2
Read the introduction to Lesson 11-2 at the top of page 583 in your textbook.
arithmetic sequence with first term 50 and common difference 9. Then add these 10 terms.
Suppose that an amphitheater can seat 50 people in the first row and that each row thereafter can seat 9 more people than the previous row. Using the vocabulary of arithmetic sequences, describe how you would find the number of people who could be seated in the first 10 rows. Sample answer: Find the first 10 terms of an
2. A triangle with sides of lengths a, a, and b is isosceles. Two triangles are cut off so that the remaining pentagon has five equal sides of length x. The value of x can be found using this equation. x2 0x x x b x x a
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Compare And Contrast Arithmetic Sequences And Series
Compare and contrast arithmetic sequences and series.
Concept Used:
A sequence is a set of numbers, called terms, arranged in some particular order.
An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference.
An arithmetic sequence is a list of terms such that any pair of successive terms has a common difference. An arithmetic series is the sum of the terms of an arithmetic sequence.
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Glencoe Algebra 2 Student Edition C2014
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College Algebra
College Algebra
Linear Algebra and Its Applications
Mathematics All Around
Essentials of Statistics, Books a la Carte Edition
Precalculus
Thinking Mathematically
Using & Understanding Mathematics: A Quantitative Reasoning Approach
A First Course in Probability