1. Lesson 1

1.9. Explore 5

Mathematics 20-1 Module 1

Module 1: Sequences and Series

 

Try This 4

 

Now that you are familiar with the variables that will be used, you are ready to construct a formula. You will do this in two different ways. After constructing your formula, you will have an opportunity to share your discoveries and your strategies with others.

 

Part 1

  1. Complete a table like Table 1 by simplifying each of the formulas. Record your answers under the Simplified Form column.

     
    TABLE 1: FORMULAS DESCRIBING SEQUENCES IN TABLE 1

    Original Formula

    Simplified Form

    1

    tn = −29 + (n − 1)(7)

     

    2

    tn = 2 + (n − 1)(3)

     

    3

    tn = n − 1

     

    4

    tn = 96 + (n − 1)(−10)

     


  2. Match each sequence in Table 2 with its corresponding formula from Table 1. For example, if you think sequence A is described by formula 3, then you would write “A3” in your copy of Table 2.

     
    TABLE 2: EXAMPLES OF ARITHMETIC SEQUENCES

     

    Sequence

    Formula from Table 1

    A

    0, 1, 2, 3, ...

     

    B

    2, 5, 8, 11, ...

     

    C

    96, 86, 76, 66, ...

     

    D

    −29, −22, −15, −8, ...

     


  3. What strategies did you use to determine how the formulas matched the sequences?

  4. What patterns do you see between the original formulas and the properties of sequences? hint

  5. Based on the pattern you observed, what would the formula to describe the following sequence look like?

     
    23, 31, 39, 47, 55, ...

  6. Study the relationship between the sequences and their corresponding simplified formulas (found in the last column of Table 2). How could you construct a formula in the simplified form without writing the formula in the original form?

Part 2

 

In Part 2, consider an arithmetic sequence that begins with the term a and has a common difference d.

 

This is a play button that opens Arithmetic Sequence Video.

Watch Arithmetic Sequence Video for an explanation of the table in question 7.



  1. Complete the table by writing an expression for t1, t2, t3, t4, t5, and t6 in terms of a and d. Simplify the expression. The first three rows have been completed for you.

     

    n

    tn

    Expression

    1

    t1

    a

    2

    t2

    a + d

    3

    t3

    a + d + d or a + 2d

     

     

     

     

     

     

     

     

     


  2. Describe the relationship between n and the coefficient of d in the expression. Express the relationship using the variable n.

  3. Extend the pattern you noticed in the table. What would be the expression for t20?

  4. What would be the expression for tn?

course folder Save your results in your course folder.

 

Compare the variables a and d.
When n = 3, what is the coefficient of d? When n = 4, what is the coefficient of d?