Lesson 3

Lesson 3: Arithmetic Series

 

Focus

 

Ever consider a career working in finance? Jobs in the finance industry involve working with money. You may know people who work with money, such as cashiers and bank tellers. However, many finance jobs do not involve actually handling bills and coins. In fact, most of the jobs in the finance industry help people or companies manage their money and make good decisions about how to spend or save their money.

 

© Alexander Raths/20754137/Fotolia

 

You may think of accounting, financial counseling, or investment banking when discussing occupations related to finance, but there are many more occupations available that are both interesting and fulfilling. For example, you could be an appraiser, controller, or risk manager.

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? Launch will help you find out.

Are You Ready?

 

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. What is an arithmetic sequence? Answer
   
   
2. Consider the arithmetic sequence 17, 21, 25, 29, ….
   
   Determine the following values.
   
   
   1. tn Answer
      
      
   2. t20 Answer
      
      
   3. n when tn = 137 Answer
      
      
3. Rearrange the general term tn = a + (n − 1)d for each of the following variables.
   
   
   1. a Answer
      
      
   2. n Answer
      
      
   3. d Answer
      
      
4. What is a system of equations? Answer
   
   
5. Solve the following system of equations.
   
   
   1. Answer
      
      
   2. Answer

How did the questions go? Depending on your answer, skip forward to Discover or work through Refresher.

Refresher

 

In the video Direct and Partial Variation: Arithmetic Sequences you will encounter the terms direct variation and partial variation. You are not required to know what these terms mean. However, in order to easily follow along, you should know that

• any linear function with a y-intercept of zero exhibits direct variation
  
  
• any linear function with a non-zero y-intercept exhibits partial variation

Discover

 

Try This 1

 

Look at the photos showing different human pyramids.

 

A: Comstock/Thinkstock, B: Photodisc/Thinkstock, C: Photodisc/Thinkstock, D: Photodisc/Thinkstock

1. Which two pyramids have people arranged by row in a way that can be described by an arithmetic sequence?
   
   
2. How many people would it take to create a 100-level pyramid?

Share 1

 

With a classmate or group, tackle these tasks.

1. Share and compare the different strategies used to find the number of people in a 100-level pyramid. What are the benefits and drawbacks of each strategy?
   
   
2. Develop a method to determine the number of people it would take to build an n-level pyramid, where n is the number of levels in the pyramid.

Save the strategy your group develops in your course folder. You will use your notes later in the lesson.

 

Explore

 

Germany’s Carl Friedrich Gauss was known as the prince of mathematicians. Gauss contributed much to the development of mathematics. Gauss was a child prodigy who had a gift for rapidly calculating complex sums in his head.

 

Gauss’ mathematical genius did not keep him from misbehaving at school. It is said that one of his teachers disciplined Gauss by assigning him what was thought to be a difficult task—calculating the sum of the first 100 natural numbers; in other words 1 + 2 + 3 + … + 100. Much to the surprise of his teachers and his classmates, Gauss calculated the sum in a matter of seconds! How did he do it?

 

As impressive as Gauss’ feat was, you don’t have to be a child prodigy to find the answer. In fact, Gauss’ problem-solving approach is simple to understand. Gauss looked at the problem from a different perspective.

 

In this lesson you will learn how Gauss saw the problem his teacher had given him. You will also learn to solve other problems involving the sum of the terms of arithmetic sequences.

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Try This 2

 

Retrieve the strategy you developed in Share 1 to find the number of people in n levels of a pyramid. How is this question similar to the problem posed to young Gauss? In Try This 2 you will see how Gauss saw the problem and how you can develop a way to solve similar problems.

 

Step 1: Print a copy of Human Pyramid.

 

Step 2: Use scissors to cut a triangle around each pyramid. Be sure to cut as close to the figures as you can without cutting off an arm or a head!

Self-Check 1

 

In Try This 2 you found the sum of all the terms in a sequence (the total number of people in each row of the pyramid). This is referred to as an arithmetic series. Use Interactive Venn Diagram to verify the similarities and differences between arithmetic sequences and arithmetic series.

Try This 3

 

In Try This 3 you will work towards finding a formula to evaluate arithmetic series.

 

Part A

 

Follow these steps to evaluate the series 1 + 4 + 7 + 10 + 13 + 16 + 19.

 

Step 1: Evaluate the series by finding the sum of the terms on your calculator.

 

Step 2: Copy the series 1 + 4 + 7 + 10 + 13 + 16 + 19.

 

Step 3: Write the same arithmetic series underneath the original series from Step 2, but with the terms in reverse order (as shown).

 

 

Step 4: Determine the sum of each column of terms. What do you notice?

 

 

Step 5: Multiply the number of columns by the sum of each column. Compare this sum with the sum you reached in Step 1.

1. What additional step should be applied to the answer in Step 5 in order to obtain the sum of the original arithmetic series?

Save your responses in your course folder.

 

Return to Discover and try using this new method to solve the problem in Try This 1. This will make a great record of your learning for this lesson.

 

Part B

 

You will use the method you used in Part A to derive a formula to find the sum of any arithmetic series.

 

Recall that the general term of an arithmetic sequence is tn = a + (n − 1)d.

2. For the arithmetic series 1 + 4 + 7 + 10 + 13 + 16 + 19, identify the parameters a and d.
   
   
3. When working with a series, think of n as the number of terms in the series. What is the value of n for this series?
   
   
4. Based on your answer from question 3, what would be the value of tn? What does tn represent in any series?
   
   
5. In the last steps of Part A, you applied the following operation to find the sum of the arithmetic series:
   
   
    
   
   
   Substitute the parameters and variables you identified in Part B questions 1 to 3 for the numbers in the above expression.

Share 2

 

With a classmate, compare the general term to the series formula. So,

 

tn = a + (n − 1)d compared to

 

Then discuss the following question:

• What does each formula represent? What patterns or connections can you see between the formulas?

Save notes from your discussion in your course folder.

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Another way to express  is .

Using the Sn Formulas to Evaluate Other Properties of Arithmetic Series

 

Besides using the two series formulas to evaluate the sum of arithmetic series, you can also use the formulas to evaluate other properties of arithmetic series. Just as you were able to use the general term tn = a + (n − 1)d to evaluate tn, a, n, and d, you can also determine all of these properties using the series formulas.

 

Example

 

Use Building an Obstacle Course to solve a problem that uses the series formulas. Pay attention to how the information provided is used to determine which formula to use.

Try This 4

 

You have seen an example of how you can use the Sn formulas to determine the properties of an arithmetic series. Apply what you have learned to the next problem, which is based on the same context as the example.

 

Goodshoot/Thinkstock

 

Problem

 

An obstacle course uses 10 log stumps. All of the logs are cut from a single log that measures 7.7 m. The difference in height between consecutive log stumps is 0.1 m. Determine the height of the shortest stump. Show your work and record each step of your solution.

 

Save your work in your course folder.

 

Share 3

1. Compare your answers with those of a classmate. There are different ways of solving this problem. Keep the following points in mind as you discuss your solutions:
   
   
   • Do your solutions match? If not, work together to identify the error(s).
     
     
   • How are your approaches different? If they are different, what are the pros and cons of each method?
     
     
2. Work with your classmate to find the height of the tallest stump. Try to find at least two approaches to solving this problem.

Save your answers in your course folder.

 

Your teacher may want to look at your work to see how your collaboration with your classmate(s) has affected your learning.

Connect

 

Lesson 3 Assignment

Lesson 3 Summary

 

In this lesson you investigated the following questions:

• How are arithmetic sequences and arithmetic series similar and different?

• How can you visualize the sum of an arithmetic series?

You extended your understanding of arithmetic sequences to arithmetic series. An arithmetic series is the sum of the terms of an arithmetic sequence. Both arithmetic sequences and arithmetic series have the property that consecutive terms are related by a common difference.

 

In general, the general term tn = a + (n − 1)d is used to determine properties of an arithmetic sequence. The formulas  and  are used to determine the properties of an arithmetic series. In the case of the general term, the parameter n refers to the position of the term; in the series formulas, n refers to the number of terms that will be added together.

 

In this lesson you learned about Gauss’ method for evaluating an arithmetic series. You visualized Gauss’ method by placing two images of pyramids together. Later in the lesson you were able to build on that initial discovery to derive formulas that could be used to determine the sum of an arithmetic series.

 

You used the formulas to solve context-based problems. You practised using the formulas with and without rearranging. You learned to match given information within a problem with the variables of each problem. You discovered that there is more than one correct way to arrive at a solution.

 

You have now completed half of Module 1. In the second half of Module 1 you will explore a different kind of sequence—geometric. Geometric sequences and series can be used to describe a variety of different phenomenon, including population growth and radioactive decay. You will learn that these sequences have many applications in finance.