An Investigation of Sequences and Series

A sequence is a function which maps the natural numbers (positive integers) as its domain onto the set of real numbers. Thus, the number of the term is the value of the independent variable evaluated in the function to produce the value of the term in the sequence. Although many sequences are possible to define, two sequences (the arithmetic and the geometric sequences) are of particular interest to us because they behave very nicely--so nicely, in fact, that we can easily write explicit definitions for them in terms of the number of the term only.

Part 1. Listed below are the first several terms in some arithmetic sequences. a: Find the difference between the values of each two consecutive terms.

Number of term 1 2 3 4 5 6 7 8 Value of term 5 8 11 14 17 20 23 26 Differences

What do you notice about the differences between the values of the consecutive terms?

Compare the number of terms in this sequence with the number of differences you found. What is their relationship? Why does this relationship exist?

______Produce a rule to predict the value of a specific term in a sequence knowing only the number of the term.

Use Y= and the table function of your calculator to verify that your rule works. b: Find the difference between the values of each two consecutive terms.

Number of term 1 2 3 4 5 6 7 8 Value of term 33 27 21 15 9 3 -3 -9 Differences

Use Y= and the table function of your calculator to verify that your rule works. c: Find the difference between the values of each two consecutive terms.

Number of term 1 2 3 4 5 6 7 8 Value of term 7 13 19 25 31 37 43 49 Differences

Use Y= and the table function of your calculator to verify that your rule works.

Now generalize your findings to describe how we might generate an explicit definition for the terms of any arithmetic sequence.

IBSL Math Methods / AP Calculus AB Type I Portfolio Assignment Page 1 of 4 Instructor: Reg Noland 6 April 2018 Part 2: Listed below are the first several terms in some geometric sequences. a: Find the ratio between the values of each two consecutive terms.

Number of term 1 2 3 4 5 6 7 8 Value of term 5 10 20 40 80 160 320 640 Ratios

What do you notice about the ratios between the values of the consecutive terms?

Compare the number of terms in this sequence with the number of ratios you found. What is their relationship? Why does this relationship exist?

______Plot the number of the term (x) versus the natural logarithm of the value of the term (y). What do you observe about the graph of this scatter plot?

Use you calculator’s Statistical Regression functions to compute an appropriate regression equation for this plot. The resulting equation should resemble ln ( y – k ) = a x + b, where k is the value of the horizontal asymptote. Exponentiate this expression with respect to e and evaluate ea and eb; then, compare the simplified exponentiated logarithmic equation with your equation above.

b: Find the ratio between the values of each two consecutive terms.

Number of term 1 2 3 4 5 6 7 8 Value of term 4374 1458 486 162 54 18 6 2 Ratios

What do you notice about the ratios between the values of the consecutive terms?

______Plot the number of the term (x) versus the natural logarithm of the value of the term (y). What do you observe about the graph of this scatter plot?

Use you calculator’s Statistical Regression functions to compute an appropriate regression equation for this plot. The resulting equation should resemble ln ( y – k ) = a x + b, where k is the value of the horizontal asymptote. Exponentiate this expression with respect to e and evaluate ea and eb; then, compare the simplified exponentiated logarithmic equation with your equation above.

IBSL Math Methods / AP Calculus AB Type I Portfolio Assignment Page 2 of 4 Instructor: Reg Noland 6 April 2018 While a sequence merely lists the values of terms generated by a sequence definition/formula, a series adds the terms in the sequence to produce a total sum or a partial sum for n terms. A series may be either finite or infinite. Finite series will have a sum, but infinite series may or may not have a sum depending upon certain conditions that we will discover. As expected, finite arithmetic and finite geometric series turn out to be nice and easily predictable.

Part 3: Let’s re-visit our first two arithmetic sequences. a: Use the rule you discovered in Part 1a and your calculator’s seq( and cumsum( functions to store the following sequence to List1, List2, and List3.

Number of term List1 1 2 3 4 5 6 7 8 Value of term List2 5 8 11 14 17 20 23 26 Cumulative Sum List3 1st Differences 2nd Differences

The cumsum( function produces the terms in a series since it adds the consecutive terms in a sequence. What do you notice about the 1st and 2nd differences between the values of the consecutive terms of the cumulative sum?

A B A + B What interesting fact do you notice about A + B?

t3 t6 What is the relation between the number of A + B

t4 t5 pairs and the number of terms? Why?

______Use the rule for the sequence and your discovery about A + B to produce a rule using  notation to predict the value of a specific term in the series knowing only the number of the term.

Use Y= and the table function of your calculator to verify that your rule works. b: Use the rule you discovered in Part 1b and your calculator’s seq( and cumsum( functions to store the following sequence to List1, List2, and List3.

Number of term List1 1 2 3 4 5 6 7 8 Value of term List2 33 27 21 15 9 3 -3 -9 Cumulative Sum List3 1st Differences 2nd Differences

A B A + B What interesting fact do you notice about A + B?

t3 t6 What is the relation between the number of A + B

t4 t5 pairs and the number of terms? Why?

______Use the rule for the sequence and your discovery about A + B to produce a rule using  notation to predict the value of a specific term in the series knowing only the number of the term.

IBSL Math Methods / AP Calculus AB Type I Portfolio Assignment Page 3 of 4 Instructor: Reg Noland 6 April 2018 Part 4: Let’s re-visit our two geometric sequences.

a: Use the rule you discovered in Part 2a and your calculator’s seq( and cumsum( functions to store the following sequence to List1, List2, and List3.

Number of term List1 1 2 3 4 5 6 7 8 Value of term List2 5 10 20 40 80 160 320 640 Cumulative Sum List3

The cumsum( function produces the terms in a series since it adds the consecutive terms in a sequence.

n t1(1- r ) S n = gives the sum of the first n terms in a geometric series where r is the common ratio and r = 1. 1- r

Use the rule you discovered in Part 2a and mathematical induction using  notation to verify this rule for the series.

b: Use the rule you discovered in Part 2b and your calculator’s seq( and cumsum( functions to store the following sequence to List1, List2, and List3.

Number of term List1 1 2 3 4 5 6 7 8 Value of term List2 4374 1458 486 162 54 18 6 2 Cumulative Sum List3

The cumsum( function produces the terms in a series since it adds the consecutive terms in a sequence.

n t1(1- r ) S n = gives the sum of the first n terms in a geometric series where r is the common ratio and r = 1. 1- r

Use the rule you discovered in Part 2a and mathematical induction using  notation to verify this rule for the series.

IBSL Math Methods / AP Calculus AB Type I Portfolio Assignment Page 4 of 4 Instructor: Reg Noland 6 April 2018