## 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

## Arithmetic Sequences And Series

An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

**Example**

2,4,6,8,10.is an arithmetic sequence with the common difference 2.

If the first term of an arithmetic sequence is *a*1 and the common difference is *d*, then the *n*th term of the sequence is given by:

$$a_=a_+d$$

An arithmetic series is the sum of an arithmetic sequence. We find the sum by adding the first, a1 and last term, an, divide by 2 in order to get the mean of the two values and then multiply by the number of values, n:

$$S_=\frac$$

**Example**

Find the sum of the following arithmetic series 1,2,3..99,100

We have a total of 100 values, hence n=100. Our first value is 1 and our last is 100. We plug these values into our formula and get:

$$S_=\frac=5050$$

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## Chapter : Series And Sequences

In this chapter well be taking a look at sequences and series. In fact, this chapter will deal almost exclusively with series. However, we also need to understand some of the basics of sequences in order to properly deal with series. We will therefore, spend a little time on sequences as well.

Series is one of those topics that many students dont find all that useful. To be honest, many students will never see series outside of their calculus class. However, series do play an important role in the field of ordinary differential equations and without series large portions of the field of partial differential equations would not be possible.

In other words, series is an important topic even if you wont ever see any of the applications. Most of the applications are beyond the scope of most Calculus courses and tend to occur in classes that many students dont take. So, as you go through this material keep in mind that these do have applications even if we wont really be covering many of them in this class.

Here is a list of topics in this chapter.

Sequences In this section we define just what we mean by sequence in a math class and give the basic notation we will use with them. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. We will also give many of the basic facts and properties well need as we work with sequences.

## Lesson 83 Analyzing Geometric Sequences And Series

**Essential Question** How can you recognize a geometric sequence from its graph?In a geometric sequence, the ratio of any term to the previous term, called the common ratio, is constant. For example, in the geometric sequence 1, 2, 4, 8, . . . , the common ratio is 2.

**EXPLORATION 1**

Recognizing Graphs of Geometric SequencesWork with a partner. Determine whether each graph shows a geometric sequence. If it does, then write a rule for the nth term of the sequence and use a spreadsheet to find the sum of the first 20 terms. What do you notice about the graph of a geometric sequence?

**EXPLORATION 2**

Finding the Sum of a Geometric SequenceWork with a partner. You can write the nth term of a geometric sequence with first term a1 and common ratio r asan = a1rn-1.So, you can write the sum Sn of the first n terms of a geometric sequence asSn = a1 + a1r + a1r2 + a1r3 + . . . +a1rn-1.Rewrite this formula by finding the difference Sn rSn and solve for Sn. Then verify your rewritten formula by funding the sums of the first 20 terms of the geometric sequences inExploration 1. Compare your answers to those you obtained using a spreadsheet.

**Communicate Your Answer**

How can you recognize a geometric sequence from its graph?Answer:

Find the sum of the terms of each geometric sequence.a. 1, 2, 4, 8, . . . , 8192b. 0.1, 0.01, 0.001, 0.0001, . . . , 10-10Answer:

**Tell whether the sequence is geometric. Explain your reasoning.**Question 1.27, 9, 3, 1, \, . . .Answer:

2, 6, 24, 120, 720, . . .Answer:

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## Sequences And Series Mathematical Practices

Mathematically proficient students consider the available tools when solving a mathematical problem.

**Monitoring Progress**

**Use a spreadsheet to help you answer the question.**Question 1.A pilot flies a plane at a speed of 500 miles per hour for 4 hours. Find the total distance flown at 30-minute intervals. Describe the pattern.Answer:

Question 2.A population of 60 rabbits increases by 25% each year for 8 years. Find the population at the end of each year. Describe the type of growth.Answer:

Question 3.An endangered population has 500 members. The population declines by 10% each decade for 80 years. Find the population at the end of each decade. Describe the type of decline.Answer:

Question 4.The top eight runners finishing a race receive cash prizes. First place receives $200, second place receives $175, third place receives $150, and so on. Find the fifth through eighth place prizes. Describe the type of decline.Answer:

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## Lesson 81 Defining And Using Sequences And Series

**Essential Question** How can you write a rule for the nth term of a sequence?A sequence is an ordered list of numbers. There can be a limited number or an infinite number of terms of a sequence.a1, a2, a3, a4, . . . , an, . . .Terms of a sequenceHere is an example. 1, 4, 7, 10, . . . , 3n-2, . . .

**EXPLORATION 1**

Writing Rules for SequencesWork with a partner. Match each sequence with its graph. The horizontal axes represent n, the position of each term in the sequence. Then write a rule for the nth term of the sequence, and use the rule to find a10.a. 1, 2.5, 4, 5.5, 7, . . .b. 8, 6.5, 5, 3.5, 2, . . .c. \d. \e. \, 1, 2, 4, 8, . . .f. 8, 4, 2, 1, \, . . .

**Communicate Your Answer**

How can you write a rule for the nth term of a sequence?Answer:nth term of a sequencean = a1 + Question 3.What do you notice about the relationship between the terms in an arithmetic sequence and a geometric sequence? Justify yourAnswer:An **arithmetic sequence** has a constant **difference between** each consecutive pair of terms. This is similar to the linear functions that have the form y=mx +b. A **geometric sequence** has a constant ratio **between** each pair of consecutive terms.

**Monitoring Progress**

**Write the first six terms of the sequence.**Question 1.

Question 27.**FINDING A PATTERN**Which rule gives the total number of squares in the nth figure of the pattern shown? Justify your answer.Answer:

**In Exercises 3138, write the series using summation notation.**Question 31.7 + 10 + 13 + 16 + 19Answer:

\Answer:

\Answer:

## Recursive Formulas For Sequences

When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way:

a_n=a_+d

The above equation is an example of a recursive equation since the nth term can only be calculated by considering the previous term in the sequence. Compare this with the equation:

a_n=a_1+d.

In this equation, one can directly calculate the nth-term of the arithmetic sequence without knowing the previous terms. Depending on how the sequence is being used, either the recursive definition or the non-recursive one might be more useful.

A recursive geometric sequence follows the formula:

a_n=r\cdot a_

An applied example of a geometric sequence involves the spread of the flu virus. Suppose each infected person will infect two more people, such that the terms follow a geometric sequence.

**The flu virus is a geometric sequence:** Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence.

Using this equation, the recursive equation for this geometric sequence is:

a_n=2 \cdot a_

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## 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:

## 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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## 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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## 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: