Center Home -> Content Areas Home -> Math Home -> Project Activities -> Excel Activities ->

Exploring and Analyzing Sequences

 Activity Description Activity Guide

• Introduce the activity by listing several sequences on the board and having the students determine the “rules” for generating such a sequence.  For this activity, the allowable rules are: 1) what to start with, and 2) what to multiply by and add on to each number.

Example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Example: 1, 3, 5, 7, 9, 11, 13, ...

Example: 1, 4, 10, 22, 46, 94, ...

• We can set up a spreadsheet to quickly generate sequences and their cumulative sequences.  Have the students create a spreadsheet like the following.  Although the formulas are given in the following figure, have the students use their knowledge of spreadsheets and formulas to create and make sense of the formulas rather than just inputting those shown.

 A B 1 SEQUENCE GENERATOR 2 1 Starting Number 3 1 Multiplier 4 1 Add On 5 6 Sequence Cumulative sequence 7 =A2 =A7 8 =A7*\$A\$3 + \$A\$4 =B7 + A8 9 10

• Fill down the formulas in cells A8 and B8 so that the first 20 numbers in the sequence and cumulative sequence can be calculated. Discuss the roles of cells A2, A3, and A4 as variables and the use of the dollar sign (\$) to indicate “absolute” cell references that do not increment when the formula is filled down.

• With the triple (1, 1, 1) as the (starting number, multiplier, add on), the counting numbers appear in column A.  Do you recognize the numbers in column B? Use a drawing or physical model to illustrate the relationship between the counting numbers and the numbers in column B (Hint: build a “growing” staircase or triangular array).

• Change the “add on” to 2.  Describe the sequence of numbers in columns A and B?  How is the sequence in related to the cumulative sequence? Again, use a physical model or drawing to illustrate the relationship (Hint: build a “growing” pattern of squares).

• Explore changing the values in the variables (starting number, multiplier, add on) and observe the resulting sequences and cumulative sequences.  Try to generate the following sequences.  Discuss the values and strategies used to generate each.  Are there more than one set of triple values that can be used to generate the same sequence?

• even numbers
• multiples of five (discuss concept of multiplication as repeated addition)
• powers of three (discuss concept of using exponents to represent repeated multiplication)
• numbers that end in seven (connect with place value concepts)
• powers of 10 (connect with place value concepts)
• In order to further analyze the sequence and cumulative sequence, create a line graph of both columns.  Explore various triples of values and analyze the results numerically and graphically.  Record any patterns and/or generalizations.

• Be sure students explore the use of negative values.  Suggest the following triple (5, -1, 5) and discuss the effects of multiplying by a negative value.  Notice the “zigzag” and “staircase” appearance in the graph.  Pose the following challenges:

• Can anyone make a zigzag without also having a staircase?
• Can anyone make a staircase without also having a zigzag?
• Can anyone make a decreasing staircase?
• Can you generalize any findings about generating a zigzag or staircase sequence?

Part 2: Exploring Convergence and Divergence in Precalculus

• Set the starting number to 1 and the add on to 0.  Explore different values for the multiplier (M).  Be sure to include positive, negative, and decimal values.  After allowing time for exploration, use the multipliers (M) in the table on the next page and record the values to which the cumulative sequences would converge (C) as n goes to infinity (n is the number of items in the sequence).  For this exploration, it may help to express the last cell in the cumulative sequences column (about the 20th term) as a proper fraction or mixed number.

 To express a value in a cell in fractional form: Click in the cell containing the value you want to express in fractional form. Under the Format menu, choose Cells... In the dialog box, click on the Number tab and choose the Fraction category.  Highlight the choice # ??/?? .  This will allow the value of that cell to be expressed as proper fraction or mixed number with a maximum of two digits in the numerator and denominator. Click on OK. (Excel does not allow values expressed as improper fractions.)

 Multiplier M Fraction Equivalent of M Cumulative sequence Convergent Value C Fraction Equivalent of C 0.5 0.25 0.75 0.3333333 0.4
• Determine a rule for the relationship between the multiplier (M) and the value to which the cumulative sequence converges (C).  Does this rule work with decimal values of M between 0 and -1?  Does the rule work with integer values of M?  Establish the conditions under which the rule applies.

• Now, fix the starting number to 1 and the multiplier to 0.5.  Try several values for the add on and observe the resultant values in the sequence.  How does the add on value effect the sequence and the cumulative sequence?

• With the starting number fixed at 1 and the multiplier at 0.5, what value should be used as the add on to make the sequence converge to 10?  One way to approach this problem is to guess and test several values.  Excel has a built in feature called Goal Seek that can also be used to solve this problem.  Use the Goal Seek to set the value in the cell of the sequence (probably in cell A25) to 10 by changing the value of the add on (cell A3).

 To use the Goal Seek: Under the Tools menu select Goal Seek... In the dialog box, fill in the boxes for Set Cell (the cell containing a formula that you want to have a specific value), To Value (the goal value for the formula), and By Changing Cell (the cell that contains the value that will effect the formula referenced in the Set Cell box). Click OK.  The spreadsheet will change the values in the “changing” cell until the “set” cell reaches the goal value. Note:  The Set Cell box must refer to a cell containing a formula that references the cell entered in the By Changing Cell box.
• Use the Goal Seek tool to find the add on value for each of the situations in the table on the next page.  Before “seeking” each value, make a hypothesis of the value needed for the add on to make the sequence converge to the intended value.
 Start Number Multiplier (M) Value Sequence Converges To (S) Hypothesized Add On Needed Actual Add On (A) 1 0.5 4 1 0.5 8 1 0.5 16 1 0.25 4 1 0.25 8 1 0.25 16
• Determine the relationship between the multiplier (M), add on (A), and the value to which the sequence converges (S)?  Express this relationship as a function of M and A.  Are there any conditions under which this relationship does not hold?

• Conjecture how the starting number would affect the value to which the sequence converges. To explore this question, fix the multiplier and add on to specific values and only change the starting number. Compare your results with your conjecture.  Discuss why and how the starting number affects the value to which the sequence converges.

• Use the sequence generator to explore the following situations.  If the situations are possible, conjecture conditions under which they can occur.

• both the sequence and cumulative sequence columns converge
• both the sequence and cumulative sequence columns diverge
• the sequence converges and cumulative sequence diverges
• the sequence diverges and the cumulative sequence converges

Discuss the results as a class.

• Create a sequence that converges to . Now create a cumulative sequence that converges to .

• Create a sequence that converges to e. Now create a cumulative sequence that converges to e.

Back to Project Activities | Back to Math Homepage