Discrete Written Report

Republic of the Philippines
DON MARIANO MARCOS MEMORIAL STATE UNIVERSITY

Mid-La Union Campus
COLLEGE OF GRADUATE STUDIES

City of San Fernando 2500, La Union
Telefax (072) 607-5798

_____________________________________________…designing change…______________

Name of Reporter: Victoria Saffire A. Lillo

Sequences & Summations

Learning Objectives:
Students will be able to…

1. Define a sequence.
2. Understand different types of sequences.
3. Use concrete strategies to determine the pattern.
4. Differentiate geometric and arithmetic sequences.
5. Recognize and apply patterns to familiar and unfamiliar situations (predictions).
6. Know that a pattern exists.
7. See patterns in life, application of patterns beyond geometric/arithmetic sequences and sum.
8. Make predictions based on the observed pattern.
9. Determine the pattern and identify what method to be used in getting the sum.
10. Investigate or discover patterns and extend them and identify a polynomial that fits a certain
sequence.

Discussion Proper
Describing Sequences

A sequence is simply an ordered list of numbers. e.g. 0, 1, 2, 3, 4, 5, . . . . We use variables to
represent terms in a sequence: 0, 1, 2, …. To refer to the entire sequence at once, we will write:

( ) ∈N or ( ) ≥ ⇒ starting with
( ) ≥ ⇒ starting with
( ) ≥ is a function with domain N where is the image of natural number n.We call a term of
the sequence. The number in the subscripts are called indices.

EXAMPLE 1
Describe the pattern, write the next term, and write a rule for the nth term of the sequence: 0, 2, 6, 12,
....
SOLUTION
You can write the terms as 0(1), 1(2), 2(3), 3(4), . . .
The next term is a5 = 4(5) = 20.
A rule for the nth term is an = (n – 1)n.

EXAMPLE 2
Describe the pattern, write the next term, and write a rule for the nth term of the sequence: – 1, – 8, – 27,
– 64, . . .
SOLUTION
You can write the terms as (– 1)3, (– 2)3, (– 3)3,
(– 4)3, . . . . The next term is a5 = (– 5)3 = – 125.
A rule for the nth term is an = (– n)3.

Two Ways to Specify Sequence:

1. Closed formula. A closed formula for a sequence ( )n∈N is a formula for using a fixed finite number
of operations on n. This is what you normally think of as a formula in n, just as if you were defining a

function in terms of n (because that is exactly what you are doing).

2. Recursive definition. A recursive definition (sometimes called an inductive definition) for a sequence
( )n∈N consists of a recurrence relation : an equation relating a term of the sequence to previous terms
(terms with smaller index) and an initial condition: a list of a few terms of the sequence (one less than the

number of terms in the recurrence relation).

EXAMPLE

Find 6 in the sequence defined by
= − − − with 0 = 3 and 1 = 4.
Solution:

Find 2, 3, 4, 5, 6. 0 = 3, 1 = 4
2 = 2 1 − 0 = 2(4) − 3 = 5
3 = 2 2 − 1 = 2(5) − 4 = 6
4 = 2 3 − 2 = 2(6) − 5 = 7
5 = 2 4 − 3 = 2(7) − 6 = 8
= 2 5 − 4 = 2(8) − 7 =

Closed Formula: = + 3

Proof:
Plug in = + to = − − − .
2 −1 − −2 = 2(( − 1) + 3) − (( − 2) + 3)

= 2( + 2) − ( + 1)

= 2 + 4 − − 1 = + 3

Common Sequences:
1. The square numbers. The sequence ( )n≥1 has closed formula = 2.

1, 4, 9, 16, 25, . . .
2. The triangular numbers. The sequence (Tn)n≥1 has closed formula = ( 2+1).

1, 3, 6, 10, 15, 21, . . .
3. The powers of 2. The sequence ( )n≥0 with closed formula = 2 .

1, 2, 4, 8, 16, 32, . . .

4. The Fibonacci numbers (or Fibonacci sequence), defined recursively by = −1 + −2 with 1 =
2 = 1.

1, 1, 2, 3, 5, 8, 13, . . .

Partial Sums:
Given any sequences ( ) ∈N, we can always form a new sequence ( ) ∈N by = 0 + 1 + ⋯ +
.
Since the terms of ( ) are the sums of the initial part of the sequence ( ) , we call ( ) the sequence
of partial sums of ( ).

EXAMPLE
1) 1 + 2 + 3 + 4 + ⋯ + 100 ⇒ ∑1 0=01

50

2) 1 + 2 + 4 + 8 + ⋯ + 250 ⇒ ∑ 2

=0


3) 6 + 10 + 14 + ⋯ + (4 − 2) ⇒ ∑(4 − 2)

=2

If we want to multiply the instead, we could write



∏

=1

For example



∏ = !

=1

Special Formulas (shortcuts)

1. ∑ =1 =
( + 1)

2. ∑ = 2

=1

2. 2 = ( + 1)(2 + 1)
6
∑

=1

EXAMPLE
Find the sum. ∑ 1 =01 2.

10 2 = 10(10 + 1)(2(10) + 1) = 10(11)(21) = 2310 = 385
6 6 6
∑

=1

Arithmetic and Geometric Sequences

Arithmetic Sequence

If the terms of a sequence differ by a constant, we say the sequence is arithmetic. If the initial term
of the sequence is and the common difference is , then we have

Recursive definition: = − + with 0 =
Closed formula: = + .

EXAMPLE

Find the recursive definition and closed formula for the arithmetic sequences. Assume the first term
is .

1) 2, 5, 8, 11, 14, …

Solution: = 5 − 2 = 3, 8 − 5 = 3

= − + ⇒ = −1 + 3 with 0 = 2
= + ⇒ = 2 + 3

2) 50, 43, 36, 29, …

Solution: = 43 − 50 = −7, 36 − 43 = −7

= − − with 0 = 50
= −

Geometric Sequence

A sequence is called geometric if the ratio between successive terms is constant. Supposed the initial
term of the sequence is and the common ratio is , then we have

Recursive definition: = − with 0 =
Closed formula: = ∗ .

EXAMPLE

Find the recursive and closed formula for the geometric sequences. Again the first term is 0.

1) 3, 6, 12, 24, 48, …

Solution: = 6 = 12 = 24 = 48 = 2

3 6 12 24

= − ⇒ = 2 −1 with 0 = 3
= ⇒ = 3 ∗ 2
2) 27, 9,3,1, 1⁄3 …

Solution: = 9 = 3 = 1 = 1/3 = 1

27 9 3 1 3

= − with 0 = 27 = ( )



Summing Arithmetic Sequences: Reverse and Add

EXAMPLE 1
Find the sum: 2 + 5 + 8 + 11 + 14 + ⋯ + 470
Solution:

= 2 + 5 + 8 + ⋯ + 467 + 470

+ = 470 + 467 + 464 + ⋯ + 5 + 2

2 = 472 + 472 + 472 + ⋯ + 472 + 472

2 = (number of terms)(472)

Determine the closed formula:

= + ⇒ = 2 + 3
470 = 2 + 3
= 156

Since 0 = 2 and 156 = 470, there are + terms, therefore there are 157 terms.

2 = (157)(472)
2 = 74,104
= 37,052

EXAMPLE 2
Find a closed formula for 6 + 10 + 14 + ⋯ + (4 − 2)
Solution:

= 6 + 10 + ⋯ + 4 − 6 + 4 − 2
+ = 4 − 2 + 4 − 6 + ⋯ + 10 + 6
2 = 4 + 4 + 4 + 4 + ⋯ + 4 + 4 + 4 + 4

2 = (number of terms)(4 + 4)

Using = 4 − 2: 2 = 4(2) − 2
2 = 6

There are − terms since there are n numbers from 1 to n, so one less if we start with 2.

2 = (n − 1)(4 + 4)
( − 1)(4 + 4)

= 2

Sum of Arithmetic Sequence Formula

When the Last Term is Given ⇒ = ( + )

2
When the Last Term is Not Given ⇒ = [2 +( −1) ]
2

Summing Geometric Sequences: Multiply, Shift & Add
Steps:

1. Multiply each term by the r(common ratio).
2. Shift over the sum to get subtraction to mostly cancel out, leaving the first & the last term.
3. Solve for S.

EXAMPLE 1

What is 3 + 6 + 12 + 24 + ⋯ + 12288?

Solution: =

= 3 + 6 + 12 + 24 + ⋯ + 12288

− 2 = 6 + 12 + 24 + ⋯ + 12288 + 24576

− = 3 + 0 + 0 + 0 + ⋯ + 0 − 24576
− = −24573
= 24573

EXAMPLE 2

Find a closed formula for :
( ) = 2 + 10 + 50 + ⋯ + 2 ∗ 5

Solution: =

= 2 + 10 + 50 + ⋯ + 2 ∗ 5

− 5 = 10 + 50 + ⋯ + 2 ∗ 5 + 2 ∗ 5 +1

−4 = 2 − 2 ∗ 5 +1

2 − 2 ∗ 5 +1
= −4

Note: The formula for the nth term of a geometric sequence whose first term is and common ratio is is
= −1.

The formula for the sum of the geometric sequence given 1, and is = (1− ) .
(1− )

Polynomial Fitting

EXAMPLE

1, 5, 14, 30, 55, …

4 , 9, 16, 25, … ⟹ First Difference
5 , 7, 9, … ⟹ Second Difference

2 , 2, … ⟹ Third Difference

We call such a sequence △ − .

1, 4, 9, 16, …

3, 5, 7 … ⟹First Difference

2, 2, … ⟹Second Difference
We call such a sequence △ − .
In general, we say a sequence is △ − if the kth differences are constant.

EXAMPLE
1) 2, 3, 7, 14, 24, 37, … △2− constant sequence
2) 1, 8, 27, 64, 125, 216, … △3− constant sequence

3) 1, 2, 4, 8, 16, 32, 64, … not △k− constant sequence for any constant k

EXAMPLE

Find a closed formula for: 0, 6, 24, 60, 120, …, assume = 0.

Solution: 0, 6, 24, 60, 120, … is △ − sequence

△ −

= 0: = + + +
0 = 0 = (0)3 + (0)2 + (0) +

=

= 1: 1 = 6 = (1)3 + (1)2 + (1) + 0

= + +

= 2: 2 = 24 = (2)3 + (2)2 + (2) + 0

= + +

= 3: 3 = 60 = (3)3 + (3)2 + (3) + 0

= + +

+ + = 6 (Eqt.1)

8 + 4 + 2 = 24 ⇒ 4 + 2 + = 12 (Eqt.2)

27 + 9 + 3 = 60 ⇒ 9 + 3 + = 20 (Eqt. 3)

Solve for a, b, c:

( . 3) − ( . 2) ⇒ 5 + = 8

( . 2) − ( . 1) ⇒ 3 + = 6

2 = 2, = 1

5 + = 8, 5(1) + = 8, = 3

+ + = 6, 1 + 3 + = 6, = 2

= + + + ⇒ = 3 + 3 2 + 2
= ( 2 + 3 + 2)
= ( + )( + )

Try this!

Determine whether the following sequences can be described by a polynomial, if so, to what degree.

1) 3, 7, 14, 24, …

2) 0, 7, 50, 183, 484, 1055, …

3) 1, 1, 2, 3, 5, 8, 13, . . .

Solving Recurrence Relations
▪ converting recursive definitions to closed formulas.

Recurrence relation
⇒ is a recursive definition without initial conditions.
Example: Fibonacci Sequence
= − + −
For the recursive definition:
= − + − = , =

EXAMPLE 1
Check that = 2 + 1 is a solution to the recurrence relation = 2 −1 − 1 with 1 = 3.
Solution
Step 1: = 1, = 2 + 1

Step 2: 1 = 21 + 1 , 1 = 3
2 −1 − 1 = 2(2 −1 + 1) − 1

= 21. 2 . 2−1 + 21. 1 − 1

= 2 + 2 − 1

= 2 + 1

∴ = 2 + 1 is a solution.

Iteration Method
▪ is based on backward substitution and observation of the pattern in the problem.

EXAMPLE
Solve = −1 + 2 , n ≥ 2, 1 = 1.
Consider = −1 + 2
∴ −1 = −2 + ( − 1)2

−2 = −3 + ( − 2)2
−3 = −4 + ( − 3)2

.
.
Thus

= −1 + 2
= −2 + ( − 1)2 + 2
= −3 + ( − 2)2 + ( − 1)2 + 2
= −4 + ( − 3)2 + ( − 2)2 + ( − 1)2 + 2

Since n ≥ 2, 1 = 1
Thus
= 1 + [22 + 32 + ⋯ + ( − 3)2 + ( − 2)2 + ( − 1)2 + 2]

= 1 + 22 + 32 + ⋯ +( − 3)2 + ( − 2)2 + ( − 1)2 + 2
( + )( + )

=

Induction

▪ is a proof technique, a style of argument to convince ourselves and others that a mathematical
statement is always true.

Steps:

1) Show true for n = 1 (Base Case)

2) Assume true for n = k Inductive Case

3) Prove/Show true for n = k + 1

4) Conclusion

EXAMPLE

Prove for each natural number n≥ 1 that 1 + 2 + ⋯ + = ( +1).

2

Proof

Let P(n) be the statement 1 + 2 + ⋯ + = ( +1).

2

We will show that P(n) is true for all natural numbers n ≥ 1.

1. Show true for n = 1 in = ( +1)
2

1(1 + 1)
1= 2

2. Assume true for n = k

1 + 2 + ⋯ + = ( + 1)

2

3. Prove true for n = k+1

1 + 2 + ⋯ + + ( + 1) = ( + 1)( + 1 + 1)

2

( + 1) ( + 1)( + 2)
2 + ( + 1) =
2

( + 1) + 2( + 1) 2 + 3 + 2
2 =2

2 + 3 + 2 2 + 3 + 2
2=
2

4. Conclusion:

By the principle of mathematical induction ( ) is true for all natural numbers ≥ 1

Worksheet # 1: Gob-Smacked Bucket Sequence

Set Your Goals
In this worksheet, you will be able to:

Objective 1: Have a positive attitude towards the completion of the worksheet.
Objective 2: Make realizable solutions to every problem.
Objective 3: Have confidence in your own ability to contribute ideas within the

group.
Objective 4: Reflect on the importance of the knowledge, skills, and values acquired

from the worksheet in dealing with life’s situations.
Objective 5: Assess yourself as part of the reflective process.

In mathematics, a sequence is a chain of numbers (or other objects) that usually follow a particular
pattern. Many professions that use mathematics are interested in one specific aspect – finding patterns,
being able to predict the future, and making decisions, that’s why it is necessary to study sequences and
summations not only because it is one of the most important chapters in discrete mathematics but it will
also help broaden the mind to solve number series and sequences. Furthermore, this will help increase the
analytical and critical thinking and our constant curiosity in looking for patterns.

This exciting worksheet is composed of a bucket of drills, performance tasks, and activities
(which are interactive) that will blow your mind. It is a sequence of challenges starting from the easiest
drill to the most difficult (performance tasks). Fundamental to our approach is a belief that learning is
most effective when students are engaged in higher-order thinking skills as they work on tasks that are
appropriately challenging for them. A task or problem is challenging if students do not know initially
how to proceed, have not been told how to do so by the teacher, and are expected to make decisions on
solutions or strategies for themselves. Of course, struggle associated with challenges can be daunting at
times and so the challenges need to be within the individual’s Zone of Proximal
Development. Completing this worksheet is a task that you should take seriously as it will test your
understanding on sequences and summations and how you can apply them in real life.

Part I: Practice Your Skill
A. Solve the following problem by answering the following questions.

Problem: Imagine a matchstick.
How many more are needed to make a square?
How many more need adding to make yet another square alongside it?
Carry on adding more squares. . .
How many matches have you added?
How many matches are there when you have made 10 squares in the row?
20 squares? 50 squares?

B. Fill in the empty boxes so that each sequence is geometric. Each change of direction is a different
sequence.

Part II: Apply Your Skill

In this section, you will solve problems accordingly. Any method will do as long as it is in
accordance with the concept of sequences and laws of mathematics. Performance task 1 can be done
individually, composed of 3 questions based on real-life which correspond to 5 points each. The score
will be based on the solutions (show all possible workings) and the final answer. Performance task 2
can be accomplished in groups since it requires deeper understanding and analysis, there are some
concepts involved other than sequence. Anent, a rubric is provided at the end of performance task 2
to ensure the coherence of the said task.

1. Performance Task 1

Problem A. People Who Confess Everyday

Parish priests hear confession every day. On the first of September, 5 people came, on the second
day, 10 people came and 15 people on the third day and so in the same increasing pattern. If the
confession is available for a month, how many people came in September?

Problem B. Hosts in a Parish

A Parish has 5 masses a day. In the first mass, there were 100 hosts during the communion. In the
second mass, the number of hosts was doubled from the last mass. This sequence continued until the
last mass. How many hosts, all in all, attended the 5 masses in a day?

Problem C. People Invited to a Wedding

My sister will be holding a wedding in a church. Each row will have 10 seats and additional two
seats for the succeeding rows. How many guests could my sister invite if the church has a total of 14
rows all in all?

2. Performance Task 2
Scenario:Mr. Dogs has asked your company(group) to design a seating plan for a new NHL arena.

Currently, his team plays in a rink like the one below.
Sample Arena

You must create a proposal to Mr. Dogs that outlines the following information. Support your
proposal with appropriate mathematics.
Here’s the arena plan.
1. Mr. Dogs wants the number of seats in the arena to be between 18 000 and 22 500. One ring of

seats around the rink is considered a row, and row 1 is considered to be the row closest to the ice.
He wants the number of seats in each row to form an arithmetic sequence, increasing by the same
number in each subsequent row. Your task is to decide on the total number of seats in the arena by
designing a seating arrangement that has a reasonable number of rows by determining:
a. The number of seats in the first row.
b. The number of rows required.
c. The number of seats by which each row increases.
d. The number of seats in the last row.
e. The total number of seats in the arena.
2. In his current arena, Mr. Dogs charges $6000 per season for seats in rows 1-10, $4000 for season
seats in rows 11-20, $3000 for season seats in rows 21-30, and $2000 for season seats in rows 31-
40. He thinks that a more fair way to decide on season ticket prices is to use a geometric sequence,
and decrease the price in each subsequent row by the same factor based on the price of the row in
front of it. For your proposal

a. Determine a reasonable price per game for each seat in the first row.
b. Determine the factor by which the cost of each seat per game will decrease in each subsequent

row from row 1.
c. Determine the price per game of each seat in the last row.

3. There are 41 home games in the regular season. Given that he needs to sell every seat in the arena
and generate at least $50 000 000 in revenue, determine the following:

a. The total revenue he will generate by selling all the seats in his rink at the prices you set above.
You may have to adjust the prices you set above to generate at least $50 000 000 in revenue.

Rubric

Level Excellent Proficient Adequate Limited Insufficient
Blank
Criteria 4 32 1
No score is awarded as
Math All required All required Some required Most required there is no evidence
Content elements are elements are elements are elements are given
present and present but may missing, or missing or
Part 1 correct contain minor contain major incorrect No score is awarded as
errors errors there is no evidence
Most required given
Math All required All required Some required elements are
Content elements are elements are elements are missing or No score is awarded as
present and present but may missing, or incorrect there is no evidence
Part 2 correct contain minor contain major given
errors errors Most required
elements are Presentation of data is
Math All required All required Some required missing or incomprehensible
Content elements are elements are elements are incorrect
Part 3 present and present but may missing, or
correct contain minor contain major Presentation of
errors errors data is vague and
inaccurate
Presents Presentation of Presentation of Presentation of
Data data is clear, data is complete data is simplistic
precise, and and unambiguous and plausible
Explains accurate
Choices Provides logical Provides Provides No explanation is
Provides explanations explanations that explanations that provided
insightful are complete but are incomplete or
explanations vague confusing.

Part III: Explore with Your Skill

These two URLs were personalized (using my account) in order to test your knowledge and
enhance your skills in sequences and summations.

URL 1 entitled Arithmetic and Geometric Sequences: Recursive and Closed Formula was made
in Quizizz (an online learning platform). This is an interactive game in which you can adjust the
settings (a timer, memes, read aloud, and power-ups). There are 15 items for multiple-choice, 2 items
for True or False, and 3 open-ended questions. For every correct answer, you will be complimented
and you will be corrected in case of an incorrect answer. This will serve as your practice, as you can
play it as many times as you want for mastery. Every time you play it, the questions and the choices
are reshuffled.

1. URL 1: https://quizizz.com/join/quiz/625a5a9526e929001e9aecad/start?studentShare=true

URL 2 entitled Sequences and Summations (Discrete Mathematics) is made in Wizer. me. This is
in the form of a worksheet that can only be done once, as when you can click “Hand In Work”, it is
automatically submitted to the creator so make you double check all your work before clicking “Hand
IN WORK”. This worksheet is composed of 4 parts, every part has directions that you need to read
carefully, so the system can check your work (tick your correct answers). Part 1 is Sorting wherein
you identify whether a sequence is an arithmetic or geometric, Part 2 is True or False, Part 3 is
Multiple-Choice and Part 4 is Problem-Solving (wherein you can upload your solution after the
question). The feedback for the first 3 parts can be seen as soon as you hand in your work but not Part
4 as it needs to be checked thoroughly.

2. URL 2: https://app.wizer.me/learn/JYV9BD

Part IV: Ponder on Your Skill
Reflect on the lesson by asking yourself the following questions.

1. Reflective question 1: Why do we need to look for patterns?

2. Reflective question 2: What is the importance of arithmetic and geometric sequences?

Appendix

Answer Key

Part I: Practice Your Skill

A. To make a square we need three more sticks.
To make a second square, we need three more matchsticks.
Notice that for each square we make, we need three matchsticks, in addition to the one we had at

the beginning. Therefore, to make 10 squares we need a total of 31 matchsticks. For 20 squares we need
61 matchsticks and for 50 squares we need 151.

The closed formula for the sum of the matchsticks is 3 + 1.

B.

Part II: Apply Your Skill

1. Performance Task 1

Problem B.

Given: =? , = 5, 1 = 5, =?, 0 = = 0
Solution: = + , = 0 + 5

30 = 5(30)

30 = 150 30
2 2
= ( 1 + ) ⇒ 30 = (5 + 150) = 2325

∴There are 2,325 people who confessed in the month of September.

Problem B.

Given: 1 = 100 , = 2, = 5 100(1−25)
(1− ) (1−2)
Solution: = (1− ) ⇒ 5 = = −3100 = 3100
−1

∴There are 3,100 attended the 5 masses.

Problem C.

Given: =? , 1 = 10, = 2, = 14, 0 = = 8
Solution: 14 = 8 + 2(14), 14 = 36

14
= 2 ( 1 + ) ⇒ 14 = 2 (10 + 36) = 322
∴My sister can invite 322 guests to her wedding.

2. Performance Task 2
Possible Solution to Arena Plan

A solution such as this could be presented in many different ways:

Number of Seats

We propose to have 460 seats in row 1 and increase the number of seats by 4 in each subsequent
row. If we have 40 rows, the total number of seats in the arena will be 21 520, as shown below.

( )Sn=néë2a + n -1 d ùû
2

= 40 éë2(460) + (40 -1)4ùû
2

= 21520

Ticket Price

We propose that the ticket price per game for seats in row 1 should be $400. Each subsequent
row should receive an 8% decrease in this price, making the ticket price per game in row 40 a
very reasonable $15.48.

tn = ar n -1

( )40-1

= 400 0.92
= 15.48

Total Revenue

Based on our proposed model, Mr. Dogs can expect total revenue of $98 868 825.80. A
spreadsheet is useful in determining the total revenue based on the above model.

Row Seats $/Gm Row's Revenue/GM Row's
1 460 Games Revenue/Season
2 464 $400.00 $184,000.00
3 468 41 $7,544,000.00
4 472 $368.00 $170,752.00 41 $7,000,832.00
5 476 41 $6,496,289.28
6 480 $338.56 $158,446.08 41 $6,027,668.07
7 484 41 $5,592,450.00
8 488 $311.48 $147,016.29 41 $5,188,289.75
9 492 41 $4,813,003.46
496 $286.56 $136,401.22 41 $4,464,557.92
10 500 41 $4,141,060.44
11 504 $263.63 $126,543.65 41 $3,840,749.39
12 508 41 $3,561,985.32
13 512 $242.54 $117,390.33 41 $3,303,242.71
14 516 41 $3,063,102.21
15 520 $223.14 $108,891.66 41 $2,840,243.43
16 524 41 $2,633,438.21
17 528 $205.29 $101,001.47 41 $2,441,544.26
18 532 41 $2,263,499.34
19 536 $188.86 $93,676.81 41 $2,098,315.73
20 540 41 $1,945,075.09
21 544 $173.76 $86,877.69 41 $1,802,923.74
22 548 41 $1,671,068.12
23 552 $159.85 $80,566.90 41 $1,548,770.69
24 556 41 $1,435,346.02
25 560 $147.07 $74,709.81 41 $1,330,157.15
26 564 41 $1,232,612.30
27 $135.30 $69,274.23 41 $1,142,161.61
41 $1,058,294.31
$124.48 $64,230.20

$114.52 $59,549.86

$105.36 $55,207.30

$96.93 $51,178.43

$89.17 $47,440.86

$82.04 $43,973.75

$75.48 $40,757.76

$69.44 $37,774.89

$63.88 $35,008.44

$58.77 $32,442.86

$54.07 $30,063.71

$49.75 $27,857.60

$45.77 $25,812.06

28 568 $42.10 $23,915.51 41 $980,535.95
$908,445.84
29 572 $38.74 $22,157.22 41 $841,614.72
$779,662.53
30 576 $35.64 $20,527.19 41 $722,236.35
$669,008.52
31 580 $32.79 $19,016.16 41 $619,674.83
$573,952.88
32 584 $30.16 $17,615.52 41 $531,580.52
$492,314.44
33 588 $27.75 $16,317.28 41 $455,928.81
$422,214.08
34 592 $25.53 $15,114.02 41 $390,975.75

35 596 $23.49 $13,998.85 41

36 600 $21.61 $12,965.38 41

37 604 $19.88 $12,007.67 41

38 608 $18.29 $11,120.21 41

39 612 $16.83 $10,297.90 41

40 616 $15.48 $9,535.99 41

Note: $/Gm =
Row's Revenue/GM = (Seats)($/Gm)

Row's Revenue/Season = (Row's Revenue/GM)(41)

Part III: Explore with Your Skill 11. 15
URL 1: 12. = −1 + 11 , 1 = 6
13. -16, -11, -6, -1
1. d = 3 14. True
2. 32, -64 15. True
3. Arithmetic 16. 19, 23, 27
4. Geometric 17. Common difference
5. r = -1/4 18. 9, 16, 23, 30, 37
6. 18 19. 204
7. 5 20. -10, -17, -24, -31, -38
8. -3, -9, -27
9. = 3(2) −1
10. = −1 − 5 , 1 = 10

URL 2:
Part 1.

Part 2.

1. TRUE 2. FALSE 3. FALSE 4. TRUE 5. TRUE

Part 3.

1. d 2. b 3. a 4. d 5. c

6. d 7. a 8. d 9. a 10. b

Part 4.

1. Solution:

4,11,20,31,44, ...
7 9 11 13 : 1st difference
2 2 2 : 2nd difference
It’s a ∆2 , use the SF of a quadratic polynomial.

= 2 + +
= 1: 1 = 4 = (1)2 + (1) +

+ + = 4 ⇒ . 1
= 2: 2 = 11 = (2)2 + (2) +

4 + 2 + = 11 ⇒ . 2
= 3: 3 = 20 = (3)2 + (3) +

9 + 3 + = 20 ⇒ . 3

Solving for a,b,c:

. 3 − 2: (9 + 3 + = 20) − (4 + 2 + = 11) ⇒ 5 + = 9

. 2 − 1: ( 4 + 2 + = 11) − ( + + = 4) ⇒ − 3 + = 7

2 = 2

= 1 5 + = 9 + + = 4

5(1) + = 9 ⇒ = 4 1 + 4 + = 4 ⇒ = −1

= 2 + + ⇒ = (1) 2 + (4) − 1
∴ = + − is the closed formula for the sequence (an)n≥1: 4,11,20,31,44, ...

2. Solution:

2 = 2 2−1 + 4 = 2 1 + 4 = 2(3) + 4 = 10
3 = 2 3−1 + 4 = 2 2 + 4 = 2(10) + 4 = 24
4 = 2 4−1 + 4 = 2 3 + 4 = 2(24) + 4 = 52
5 = 2 5−1 + 4 = 2 4 + 4 = 2(52) + 4 = 108

∴3,10,24,52,108…

To find 0:

3 = 2 0 + 4 1
2
3 − 4 = 2 ⇒ 0 = −

Then find a recursive definition for the sequence 10, 24, 52, 108, . . .

∴ = − + with = −



3. Solution:

Let P(n) be 12 + 22+…+ 2 = ( +1)(2 +1) for ∈ ℕ.
6
1(1+1)(2(1)+1)
a. For n=1: (1)2 = 6 True

b. Assume n=k: 12 + 22+…+ 2 = ( +1)(2 +1) is true.
6

c. Prove n=k + 1: 12 + 22+…+ 2 + ( + 1)2 = +1( +1)+1)(2( +1)+1)
6

( +1)6(2 +1)+( + 1)2 = ( +1)( +2)(2 +3)
6

2 3 +63 + +( 2 + 2 + 1) = 2 3+9 2+13 +6
6

2 3 + 3 2 + + 6 2 + 12 + 6 2 3 + 9 2 + 13 + 6
6 =6

2 3 + 3 2 + + 6 2 + 12 + 6 2 3 + 9 2 + 13 + 6
6 =6

2 3 + 9 2 + 13 + 6 2 3 + 9 2 + 13 + 6
6=6

n = k + 1 is true

d. P(n) is true for ∈ .

4. Solution:

Let P(n) be 13 + 23 + 33+. . . + 3 = 2( +1)2 is true for ∈ ℕ.
4

a. For n = 1: (1)3 = 12(1+1)2 = 1 True
4

b. Assume n = k : 13 + 23 + 33+. . . + 3 = 2( +1)2 is true.

4

c. Prove n = k + 1: 13 + 23 + 33+. . . + 3 + ( + 1)3 = ( +1)2(( +1)+1)2
4

2( + 1)2 + 4( + 1)3 ( + 1)2( + 2)2
4 =4

2( 2 + 2 + 1) + 4( 3 + 3 2 + 3 + 1) ( 2 + 2 + 1)( 2 + 4 + 4)
4 =4

4 + 2 3 + 2 + 4 3 + 12 2 + 12 + 4 4 + 6 3 + 13 2 + 12 + 4
4 =4

4 + 6 3 + 13 2 + 12 + 4 4 + 6 3 + 13 2 + 12 + 4
4=4

n = k + 1 is true
d. P(n) is true for ∈ .

5. Solution:

Find the sum: 5 + 9 + 13 + 17 + 21 + ⋯ + 533
Solution:

= 5 + 9 + 13 + ⋯ + 529 + 533

+ = 533 + 529 + 525 + ⋯ + 9 + 5
2 = 538 + 538 + 538 + ⋯ + 538 + 538

2 = (number of terms)(538)

Determine the closed formula:

= + ⇒ = 5 + 4
533 = 5 + 4
= 132

Since 1 = 5 and 132 = 533, there are terms, therefore there are 132 terms.

2 = (132)(538)
2 = 71016
= 35508

Part IV: Ponder on Your Skill

Reflective question 1: (POSSIBLE ANSWER)
Finding patterns is extremely important. To find patterns among problems, we look for things that

are the same (or very similar) in each problem. Patterns exist among different problems and within
individual problems. We need to look for both. In sequence, if we can identify the pattern of numbers, so
then we will be able to determine what type it is and establish what formula/method to be used. In short,
patterns make our tasks simpler. Problems are easier to solve when they share patterns because we can use
the same problem-solving solution wherever the pattern exists.

The more patterns we can find, the easier and quicker our overall task of problem solving will be.

Reflective question 2: (POSSIBLE ANSWER)
There are different types of sequences and each of these has different roles and objectives in

describing things mathematically. One of these is the arithmetic sequence. The arithmetic sequence or
sometimes called arithmetic progression is a sequence of numbers in which each consecutive terms have a
common difference. This concept is widely used in different aspects of life. The arithmetic sequence is
important in real life because this enables us to understand things with the use of patterns. An arithmetic
sequence is a great foundation in describing several things like time which has a common difference of 1
hour. An arithmetic sequence is also important in simulating systematic events. Through the arithmetic
sequence, we are able to create ideas and live by applying systematic knowledge.

Another is a geometric sequence. Such a sequence is defined as the next entry is the ratio multiplied
by the previous entry. An application of it is: A population growth in which each person decides not to have
another child based on the current population then population growth each year is geometric. Geometric
series is useful because it can be used as a model of real-life situations which can find its application in
physics, banking, and finance, etc.

Anent, arithmetic and geometric sequences are ubiquitous. They are all around, so they matter to
us and that’s why they are really important.