summation notation4

8/9/2019 Summation Notation4

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Summation Notation

and Its Properties

Summation Notation

and Its Properties


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and Its Propertiesand Its Properties

At the end of the 60-minute period,at least 85% of the students should beable to:

At the end of the 60-minute period,at least 85% of the students should beable to:

Define and expand summations

Write a sum in summation form

Evaluate simple summations bydefinition

Identify the different parts of asummation

Define and expand summations

Write a sum in summation form

Evaluate simple summations bydefinition

Identify the different parts of asummation


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CHALLENGE!!!CHALLENGE!!!

An online shopping site reduced the price ofone computer model by 25% for a sale. By

what percentage of the sales price must itbe increased to put the computer modelback at its original price?

An online shopping site reduced the price ofone computer model by 25% for a sale. By

what percentage of the sales price must itbe increased to put the computer modelback at its original price?

Suppose the computer originally cost $100. Withthe 25% deduction, it would cost $75. To bringthe price back to the original $100, youd have toadd $25, which is 1/3 or $75 or 33.3%


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MENTALCOMPUTATIONMENTALCOMPUTATION

1. 2 x 3 6 1

2. 21 7 1 x 4

3. 5 + 9 x 2 11

4. 8 10 5 + 135. 7 7 7 x 7

1. 2 x 3 6 1

2. 21 7 1 x 4

3. 5 + 9 x 2 11

4. 8 10 5 + 135. 7 7 7 x 7


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AnswersAnswers

1. 2 x 3 6 1

2. 21 7 1 x 43. 5 + 9 x 2 11

4. 8 10 5 + 13

5. 7 7 7 x 7

1. 2 x 3 6 1

2. 21 7 1 x 43. 5 + 9 x 2 11

4. 8 10 5 + 13

5. 7 7 7 x 7

1. 0

2. -1

3. 12

4. 19

5. -48


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SUMMATION NOTATIONSUMMATION NOTATION

A convenient and simple form ofshorthand used to give a conciseexpression for a sum of the values of

a variable.

A convenient and simple form ofshorthand used to give a conciseexpression for a sum of the values of

a variable.

Also known as sigma notation


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ExamplesExamples

1.

2.


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3. What if in Example 1,

thenandxxx 7,2,5 321!!!


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4,9,6,1,1,3 654321

!!!!!!

yyyyyy

4. What if in Example 2,


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y6 + y7 + y8 + y9 + . . . + y15y6 + y7 + y8 + y9 + . . . + y15

!!

15

6ii

5. Write the following expansion insummation form:


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z2 + z3 + z4 + z5 + y2+ y3+ y4 + y5z2 + z3 + z4 + z5 + y2+ y3+ y4 + y5

!!

5

2

5

2 ii

ii yz

T

here are two ways to write the sum above insummation form. These are:

or )(

5

2i

ii y!

6.Write the following expansion

in summation form:


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!!

100

1i

i

7.Write the sum 1+2+3++100

in summation form:


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Using the examples as your guide,

1. How do you write the sum ofexpressions in summation form?

2. What are the important thingsthat you have to consider inwriting a sum in summation form?

Using the examples as your guide,

1. How do you write the sum ofexpressions in summation form?

2. What are the important thingsthat you have to consider inwriting a sum in summation form?

Consider: v15 + v16 + v17 + . . . + v35


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Summation

sign

Typical element

Stopping point or upper limit of summation

Index ofsummation

Starting point or lowerlimit of summation

Parts of a summation notation

vi

n

i =1


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Q: Guess which sum isgreater. Then check bydoing the addition with a

pencil or a calculator.Don't forget to add thenumbers according toplace value!

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9

8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8

76

54

3 2 1 1 2 34

56

76 5 4 3 2 1 1 2 3 4 5 6

5 4 3 2 1 1 2 3 4 5

4 3 2 1 1 2 3 4

3 2 1 1 2 3

2 1 1 2

1 1

A: You may be surprisedwhen you find out that thesums are exactly the same!They both equal1,083,676,269.


9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9

8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8

7 6 5 4 3 2 1 1 2 3 4 5 6 7

6 5 4 3 2 1 1 2 3 4 5 6

5 4 3 2 1 1 2 3 4 5

4 3 2 1 1 2 3 4

3 2 1 1 2 3

2 1 1 2

1 1

9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9

8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8

76

54

3 2 1 1 2 34

56

76 5 4 3 2 1 1 2 3 4 5 6

5 4 3 2 1 1 2 3 4 5

4 3 2 1 1 2 3 4

3 2 1 1 2 3

2 1 1 2

1 1


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1. 6 x 2 4 -3 +1

2. 11 + 5 8 2 x 4

3. 10 6 + 5 x 3 3

4. 15 5 x 6 +7 - 85. 28 7 x 4 9 + 9

1. 6 x 2 4 -3 +1

2. 11 + 5 8 2 x 4

3. 10 6 + 5 x 3 3

4. 15 5 x 6 +7 - 85. 28 7 x 4 9 + 9


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AnswersAnswers

1. 6 x 2 4 -3 +1

2. 11 + 5 8 2 x 4

3. 10 6 + 5 x 3 3

4. 15 5 x 6 +7 - 85. 28 7 x 4 9 + 9

1. 6 x 2 4 -3 +1

2. 11 + 5 8 2 x 4

3. 10 6 + 5 x 3 3

4. 15 5 x 6 +7 - 85. 28 7 x 4 9 + 9

1. 1

2. 0

3. 9

4. 175. 16


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Walking DictionaryQ: Delphinia is a walking dictionary. She

isn't satisfied unless she's showing off herhuge vocabulary. Here's how Delphinia saysthese four well-known proverbs. See if youcan figure out how most of us would say

them:

1. The previously arriving feathered,winged biped captures the invertebratecrawler.

2. It's vastly preferable to remainunscathed than to be forced intorepentance.

3. If you refrain from interrogating me,I'll articulate to you no prevarications.4. When the terrain becomes arduous,

the resolute become vigorous.

A:1. The early bird catches the worm.2. Better safe than sorry.3. Ask me no questions, I'll tell you no

lies.

4. When the going gets tough, thetough gets going.


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2. 16 x (2) + 8 (4) (10)3. 48 6 (1) - (25) + 35. (14 x 2 + 9)/ [(7) 7]4. 9 x 9 + 9 (9) ( 9)1. 6 + 9 3(-2)


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2. -24

AnswersAnswers

3. 20

5. undefined4. 89

1. 0

5. (14 x 2 + 9)/ [(7) 7]

1. 6 + 9 3(-2)

4. 9 x 9 + 9 (9) ( 9)

3. 48 6 (1) - (25) + 3

2. 16 x (2) + 8 (4) (10)


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x1 = 5 x2 = 3 x3 = 7 x4= 4 x5 = 8 x6= 6

y1 = 3 y2 =9 y3 =1 y4= 10 y5 = 11 y6= 2


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AnswersAnswers

=7

=14

=21

=8

=16

=15


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AnswersAnswers

=30

=2

=35

=24

=25

=35

=40

=45


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Using your answers in Part I as basis, answer thefollowing questions:

1.What can you say about your answers in numbers 1 to 5?Are they related in some way? If yes, how are theyrelated?

2. Can you suggest an alternative solution for numbers 1 to 5?

3. For numbers 6 to 8, do you observe any association betweenthem and items 1 and 4? What is it?

4. Is there an easier and faster way of solving the summationsin numbers 9 and 10? If yes, how do you do it?

5. How about numbers 11 to 14, can you figure out thesimplest (and the fastest!) way of solving them? (Hint:You need to consider the upper and lower limits.)


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AssignmentAssignment

Do pages 65-66 of the textbook.

Copy and answer on your notebook.Show all solutions.

1. Write each of the following expressions in full:

2. In x1=3, x2=4, x3=5, y1= 3, y2=3, y3= 23. evaluate the following:


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AssignmentAssignment

3. Show that a=mna, a is constant.

4. Let x1=2, x2=8, x3=9, x4=10, and x5=15,

5. Let x1=13, x2=8, x3=12, y1=3, y2=6, y3=15,compute the following:


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Q: Palindrome is a word (or phrase or sentence) thatreads the same from left to right as it does from right toleft. For example, a palindrome that means "trick or joke"

is gag. For each of the definitions below, suggest apalindrome.

1. Midday2. Title for a lady

3. Tier4. Sound a chick makes5. Part of the human body

6. Something that fails to work7. An entertainer's short-term job8. Sound a horn makes


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AnswersAnswers

1. noon2. madam

3. level4. peep5. eye

6. dud7. gig8. toot


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1. 5 x 8 4 3 x 72. 2 x 6 3 + (9) x 23. 1100 24. 3 x 2 [13 x 100 50 26]5. [14 (2 + 5) + 2] 3


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AnswersAnswers

1. 5 x 8 4 3 x 7

2. 2 x 6 3 + (9) x 23. 1100 24. 3 x 2 [13 x 100 50 26]5. [14 (2 + 5) + 2] 3

1.11

2.223.0.54.Undefined5.0


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1. Evaluate

ExamplesExamples

= (n+1)(n/2)

= 77(78)/2= 125(126)/2= 51(50/2)= 1275

2. Evaluate

= 3003

3. Evaluate

= 7875


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ExamplesExamples

1. Evaluate = [(n/2)(n+1)]2

= [(4/2)(4+1)]2

= [2(5)]2

= (10)2

= 100

2. Evaluate

= [(25/2)(25+1)]2

= [13(25)]2

= 105625

3. Evaluate

= [20(21)/2]2

= 44100


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ExamplesExamples

1. Evaluate = n(n+1)(2n+1)6

= 5(6)(11)6

= 55

2. Evaluate

=13(14)(27)6

= 819


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Q: Mary is twice as old asher brother and half as old asher father. In 22 years, herbrother will be half as old as

his father. How old is Marynow?

A: Mary is now 22 years old.


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1. 5/7 - 32. -2/9 + 53. 4 + 3/44. 1/5 - 25. 10/15 + 1


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AnswersAnswers

1. -16/7

2. 43/93. 19/44. -9/5

5. 5/3


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Summation Equivalent/Formula

Summation Equivalent/Formula


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x1 = 4 x2 = 3 x3 = 8 x4= 7 x5 = 16 x6= 10y1 = 3 y2 = 9 y3 = 11 y4= 5 y5 = 7 y6= 6


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x1 = 4 x2 = 3 x3 = 8 x4= 7 x5 = 16 x6= 10y1 = 3 y2 = 9 y3 = 11 y4= 5 y5 = 7 y6= 6


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AssignmentAssignment(1/2 sheet of paper)

1.

2.

3.

4.

5.

summation notation4

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