Wednesday, September 20, 2023

Algebra 2 Sequences And Series

Sequences And Series Cumulative Assessment

Arithmetic Sequences & Series (Learn Algebra 2)

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 a1 and the common difference is d, then the nth 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.

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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Big Ideas Math Algebra 2 Answers Chapter 8 Sequences And Series

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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: