SHS

Pre-Calculus
Quarter 1 – Week 8
Pre-Calculus
Grade 11 Quarter 1 – Week 8
First Edition, 2020

Copyright © 2020
La Union Schools Division
Region I

All rights reserved. No part of this module may be reproduced in any form without
written permission from the copyright owners.

Development Team of the Module

Author: Josefina L. Estela, MT II

Editor: SDO La Union, Learning Resource Quality Assurance Team
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Schools Division Superintendent
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Michael Jason D. Morales, PDO II
Claire P. Toluyen, Librarian II
Pre-Calculus
Quarter 1 – Week 8
 Target

From the previous lesson, you have learned how to illustrate a series. Now,
we will represent these series such that we will not be writing the sum in a long
method. The use of sigma notation will shorten the series and make it more
compact in form. It will help us shorten in writing out the long sum of a sequence.
The sigma notation will be used to denote a sum.

After going through this module, you are expected to:

1. Use the sigma notation to represent a series. (STEM_PC11SMI-Ih-3)

Before going on, check how much you know about this topic. Answer the
pretest below in a separate sheet of paper.

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 Module Writing Sums in Sigma Notation

n
∑3i
i =1

Figure 1 shows the sigma notation. The uppercase Greek letter (sigma) is used to
indicate a “sum.” The notation consists of several components or parts.

 The summation symbol (∑), which is the Greek upper-case letter. The
summation symbol, ∑, instructs us to sum the elements of a sequence. A
typical element of the sequence which is being summed appears to the right
of the summation sign.

 The variable of summation, i.e. the variable which is being summed

The variable of summation is represented by an index which is placed
beneath the summation sign. The index of summation is often represented
by i. (Other common possibilities for representation of the index are j and t.)
The index appears as the expression i = 1. The index assumes values
starting with the value on the right hand side of the equation and ending
with the value above the summation sign.

 The starting point for the summation or the lower limit of the summation

 The stopping point for the summation or the upper limit of summation

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 Jumpstart

For you to understand the lesson well, let us learn from this
story. Have fun and good luck!

In Module 7, we introduced the concept of summation. Carl Friedrich Gauss (1777-

1855) was a German mathematician who contributed in many fields of
mathematics and science and is touted as one of history's most influential
mathematicians.

I love the story of Carl Friedrich Gauss—who, as an

elementary student in the late 1700s, amazed his teacher
with how quickly he found the sum of the integers from 1 to
100 to be 5,050. Gauss recognized he had fifty pairs of
numbers when he added the first and last number in the
series, the second and second-last number in the series, and
so on.
As the story goes, when Gauss was a young boy, he was
given the problem to add the integers from 1 to 100.
Remember that there were no calculators in those days!
As the other students struggled with this lengthy addition problem, Gauss saw a
different way to attack this problem. He listed the first 50 terms, and then listed
the second fifty terms in reverse order beneath the first set. You can think of it as
he "wrapped" the series back onto itself.

For example: (1 + 100), (2 + 99), (3 + 98), . . ., and each pair has a sum of 101.

50 pairs × 101 (the sum of each pair) = 5,050.

Gauss them added the paired values, noticing that the sums were all the same
value (101). Since he had 50 such pairs, he multiplied 101 times 50 and obtained
the sum of the integers from 1 to 100 to be 5050.

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 Pretest

Directions: Read carefully each item. Use separate sheet for your answers.
Write only the letter of the best answer for each test item.

1. In the expression 肈ሞ , which is the index of summation?

A. 8 C. 3
B. 5 D. 2
. In the expression 肈
ሞ , which is the upper limit?
A. 8 C. 3
B. 5 D. 2
ሞ. Rewrite the series 16 + 25 + 36 + 49 + 64 + 81 using sigma notation
A. 肈h
1 B. 肈1 C. 肈1 D. 肈h

h. Using the series 4 + 5 + 6 + 7 + 8 + 9, express in sigma notation

A. 肈h B. 肈1 C. 肈h D. 肈1

. The series 4 + 16 + 24 + 256 + 1024 can be expressed as _____.

A. 肈 B. 肈
h C. 肈1
h D. 肈h

. Rewrite the expression 肈1 1 as a sum.

A. 4 + 19 + 44 + 79 + 124 C. 4 + 16 + 64 + 256 + 1024
B. 19 + 44 + 79 + 124 + 256 D. 4 + 19 + 44 + 64 + 79 + 124

7. The expression 肈1 can be written as _____.

A. 3 + 8 + 15 + 24 + 35 + 48 C. 3 + 8 + 15 + 34 + 45 + 48
B. 1 + 2 + 3 + 4 + 5 + 6 D. 3 + 8 + 11 + 15 + 18 + 24

. Express 肈1 h as a sum.
A. 9 + 10 + 11 + 12 + 13 + 14 + 15 C. 9 + 24 + 49 + 84 + 129 + 184
B. 9 + 24 + 49 + 84 + 175 D. 24 + 49 + 84 + 129 + 184 + 204

. The expression 肈1 1 is equivalent to _____.

A. 3 + 8 + 11 + 14 + 17 C. 3 + 9 + 19 + 33 + 73
B. 9 + 19 + 33 + 51 + 73 D. 3 + 9 + 19 + 33 + 51

10. Rewrite the expression 肈0

h0 as a series.
A. 40 + 39 + 36 + 31 + 15 + 4 C. 39 + 36 + 31 + 24 + 15 + 4
B. 40 + 39 + 36 + 31 + 24 + 15 D. 39 + 4

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.

Discover

Let's generalize what Gauss actually did. How did he do it? Let us discover the
pattern by filling up the table.

Guided Activity
Task/Question Answer/Notation
1. Using the subscripted a notation,
how do you represent the first term,
second term and third term
2. Represent the last term.
3. Represent the sum of all integers
from 1 to 100 using the subscripted
a notation
4. What is the sum of the wrapped
pairs in symbol?
(Hint: All of Gauss' wrapped pairs have
a sum of 101. While any pair could be
used, it will be easier if we choose the
pair of first and last terms, as these
terms are usually more readily
available.)
5. How many pairs are there? What is
the relation of these pairs to the
number of terms n?
6. Gauss multiplied this sum (your
answer in number 4) times the
number of pairs (which is HALF the
number of terms in his sequence.)
Can you give the product? Is this
starting to look familiar?
7. What formula did we arrive at?
8. Rewrite this formula using sigma
notation

100
So, the sum of 1 + 2 + 3 + … + 98 + 99 + 100 can be written as 肈1

In this case, sigma notation is a concise and convenient way to represent long
sums. Suppose that we are given a long sum and we want to express it in sigma
notation. How should we do this?

Let us take the two sums we started with. If we want to write the sum
1+2+3+4+5
in sigma notation, we notice that the general term is just k and that there are 5
terms, so we would write

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 1+2+3+4+5= 肈1
To write the second sum
1 + 4 + 9 + 16 + 25 + 36
in sigma notation, we notice that the general term is k2 and that there are 6 terms,
so we would write
1 + 4 + 9 + 16 + 25 + 36 = 肈1

Here are more examples:

Problem: Solution:

1. Use sigma notation to Look for a pattern positio

represent based upon the term
n
4 + 8 + 12 + 16 + ... position of each
for the first 12 terms. term. Often making 1 4 Possible answer:
1
a table will let the
pattern to be more 2 8 h
easily seen. 3 12
肈1
Sequence formula:
ak = 4k 4 16

2. Use sigma notation to Again, look for a positio

represent pattern. Remember term
n Possible answer:
-3 + 6 - 9 + 12 - 15 + ... the concept
for the first 50 terms. regarding using 1 -3
0
powers of (-1) to
affect the signs of 2 6 1 ሞ
the terms. 3 -9
肈1
Sequence formula:
an = (-1)n•3n 4 12

3. Express as a sum: Notice that the starting value is i = 2. While the starting
h value is usually 1, it can actually be any integer value.
Also notice how ONLY the variable i is replaced with the
肈 values 2, 3, and 4:
h

肈 ሞ ሞ h h

4. Express as a sum: Yes, it is possible to calculate a summation on an

1 expression starting with a negative number. Substitute -
h 1 2, -1, 0 and 1. Remember, however, that when working
肈 with sequences, the lowest starting value is 1.
1

h 1 肈 h 1 h 1 1 h 0 1
肈
h 1 1

5. Jhometrie is starting a An increase of 10%, is equivalent to 110% per week in

6 week jogging program. the number of miles.
She will jog 8 kilometers Week 1: 8 km
the first week and Week 2: 8 + .10(8) or 1.10(8) km

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increase the distance by Week 3: 1.10(1.10)(8) = (1.10)2(8)
10% per week. Using Week 4: 1.10(1.10)(1.10) (8) = (1.10)3(8)
sigma notation, write an (and so on ...) The pattern is (1.10)n-1(8).
expression to represent
the total number of
kilometers she will have 1.10 1
jogged over the 6 week 肈1
program.

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 Explore

Here are some enrichment activities for you to work on to master

and strengthen the basic concepts you have learned from this

Enrichment Activity 1: I LIKE IT SHORTEN!

Rewrite each series using sigma notation.
1. 0 + 4 + 8 + 12 + 16 Answer: _______________
2. 601 + 602 + 603 + 604 + 605 Answer: _______________
h
3. ሞ 7
1 Answer: _______________
1 ሞ
4. 0 ሞ h
Answer: _______________
5. 1 + 4 + 9 + 16 + 25 + 36 Answer: _______________
10
6. 10 ሞ
Answer: _______________
7. 3 + 9 + 27 + 81 + 243 Answer: _______________
1 1 1 1 1 1
8. h 1 ሞ h
Answer: _______________
1 ሞ h
9. 0 ሞ h
Answer: _______________
10. 0 + 1 + 2 + 3 + 4 + 5 Answer: _______________

Enrichment Activity 2: TRANSFORM ME!

Rewrite each series as a sum.
1. 肈
00 Answer: ___________________________________
2. 肈
ሞ0 Answer: ___________________________________
7
3. 肈ሞ
ሞ0 Answer: ___________________________________
4. 肈1 1
Answer: ___________________________________
5. 肈
100 Answer: ___________________________________
ሞ
6. 肈0
Answer: ___________________________________
7. 肈
Answer: ___________________________________
10
8. 肈
1 Answer: ___________________________________
h 70
9. 肈1
Answer: ___________________________________
h
10. 肈0
ሞ Answer: ___________________________________

Great job! You have understood the lesson. Are

you now ready to summarize?

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 Deepen

Think Outside the Box!

1. Rewrite the following so that it starts at x = 0

10

1
肈7

2. Are these equal? Why or why not?

0 70
1 1
h
0
肈1 肈 1

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 Gauge

Directions: Read carefully each item. Use separate sheet for your answers.
Write only the letter of the best answer for each test item.

1. In the expression 肈 ሞ , which is the index of summation?

A. 8 C. 3
B. 5 D. 2
2. In the expression 肈
ሞ , which is the upper limit?
A. 8 C. 3
B. 5 D. 2
3. Rewrite the series 20 + 25 + 30 + 35 + 40 using sigma notation.
A. 肈h
h B. 肈h
C. 肈1 D. 肈h

4. Using the series 0 + 1 + 2 + 3 + 4 + 5, express in sigma notation.

A. 肈1 B. 肈0
C. 肈0 D. 肈1

5. The series 15 + 18 + 21 + 24 + 27 + 30 can be expressed as _____.

A. 肈ሞ B. 肈 ሞ C. 10肈 ሞ D. 肈

6. Rewrite the expression 肈1 as a sum.

A. 4 + 10 + 20 + 34 + 52 C. 20 + 34 + 52 + 74 + 96
B. 10 + 20 + 34 + 52 + 74 D. 4 + 52 + 74 + 96 + 106

7. The expression 肈1 1 can be written as _____.

A. 2 + 6 + 12 + 20 + 28 + 30 C. 0 + 2 + 6 + 12 + 20 + 30
B. 0 + 2 + 6 + 12 + 18 + 20 D. 0 + 30

8. Express 肈1 ሞ h as a sum.
A. 31 + 52 + 79 + 112 + 154 C. 7 + 16 + 31 + 52 + 112 + 184
B. 7 + 16 + 31 + 52 + 79 + 112 D. 31 + 52 + 79 + 112 + 151 + 204

9. The expression h肈1 ሞ 1 is equivalent to _____.

A. 2 + 11 + 26 + 41 C. 2 + 11 + 26 + 44
B. 2 + 11 + 26 + 43 D. 2 + 11 + 26 + 47

10. Rewrite the expression 肈0 0 as a series.

A. 19 + 16 + 11 + 4 + (-3) + ( -16) C. 19 + 16 + 11 + 4 + (-5) + ( -16)
B. 19 + 16 + 11 + 4 + (-5) + ( -18) D. 19 + 16 + 10 + 4 + (-5) + ( -16)

Great job! You are almost done with this module.

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References

Printed Materials:
Department of Education. (2016). Unit 2: Precalculus, Teacher’s Guide (pp.
87-90). Pasig City, Philippines
Garces, Ian June L. et al. (2016). Pre-Calculus. Manila, Philippines: Vibal
Group, Inc.

Website:
Gauss on Arithmetic Sequences. Retrieved July 24 from
https://mathbitsnotebook.com/Algebra2/Sequences/SSGauss.html
Sigma Notation. Retrieved July 23, 2020 from
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-sigma-2009-1.pdf
Sigma Notation and Series. Retrieved July 27, 2020 from
https://mathbitsnotebook.com/Algebra2/Sequences/SSsigma.html

Summation Notation Worksheet . Retrieved July 23, 2020 from

http://www.web.pdx.edu/~sstrand/Math253/HW/SigmaNotationWS.pdf
Summation Notation + Work With Sequences. Retrieved July 23, 2020 from
http://math.wsu.edu/faculty/cjacobs/201/sec1.5suppl.pdf
The Story of Gauss. Retrieved July 24, 2020 from
https://www.nctm.org/Publications/Teaching-Children-
Mathematics/Blog/The-Story-of-Gauss/

Software:
Kuta Software LLC Infinite Algebra 2 (Trial Version)

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