PRE-CALCULUS

QUARTER 1, WEEKS 9

Name: ________________________________ Date: ________________

Grade & Section: ______________________

SIGMA NOTATION

I. Learning Competency
Use the sigma notation to represent a series (STEM_PC11SMI-Ih- 3)
Apply the use of sigma notation in finding sums.
illustrate

II. Objectives
At the end of the lesson, the student should be able to:
1. write a series in sigma notation or summation notation ;
2. evaluate sums written in sigma notation;
3. determine the properties of sigma notation; and
4. calculate sums using sigma notation.

III. Introduction (Key Concept)

For convenience of writing a series, summation or sigma notation can be used.
The series 2 + 4 + 6 + 8 + 10 can be written in more concise way. It can be expressed as
5

 2i . The expression is read “the summation from 1 to 5 of 2i”. The number of terms of the
i 1

series is the difference between the upper bound and the lower bound plus one, (5 -1 =
4+1 = 5). Generally, we have

Type equation here.

1
Evaluate a sum in sigma notation

The terms of the series above are generated by successively replacing the index of
summation with the consecutive integers from the first up to the last values of n. Thus, the
values of n for the series above are 1, 2, 3, … up to 5.

Illustrative Examples.

1. State the index and the number of terms in the following series
10 15 5
a.  3n
n 1
b.  3  2k
k 3
c. 3  2
j 3
j 1

Solutions:
a. Index is n
# of terms 10-1 = 9 +1 = 10
b. Index is k
# of terms 15-3 = 12 +1 = 13
c. Index is j
# of terms 5-3 = 2 +1 = 3

5
2. Write  3n  4 in expanded form and find the sum
n 1
Solution:
5

 3n  4 = [3(1) − 4] + [3(2) − 4] + [3(3) − 4] + [3(4) − 4] + [3(5) − 4]

n 1
= (3 − 4) + (6 − 4) + (9 − 4) + (12 − 4) + (15 − 4)
= −1 + 2 + 5 + 8 + 11
= 25
3. Use sigma notation to express 3 + 10 + 17 + 24 + 31.
Solution:
Search for a pattern
3 + 10 + 17 + 24 + 31
= 3 + 0(7) + 3 + 1(7) + 3 + 2(7) + 3+ 3(7) + 3 + 4(7)

If n is replaced by 1, 2, 3, 4, 5, the pattern is 3 + (n-1)7, therefore the series

can be expressed as
5 5

 3  (n  1)7   7n  4
n 1 n 1

Properties of sigma notation

Generally, sigma notation looks like

n

 f (i)  f (m)  f (m  1)  ....  f (n)

im

Similarly, just like in our previous discussion, I is the index of summation, m is the
lower bound, n is the upper bound and f(i) is a term. The sum of the first term o the
sequence with nth term an is

2
 n

a
i 1
i  a1  a2  ...  an

To evaluate the sum of a series using sigma notation, we have to consider the
properties of sigma notation.

n n n

1.  ai  bi   ai   bi
i m i m i m
n n
2.  ca
im
i  c  ai
i m
n n
3.  c   c(n  m  1)
i m i m
n n
4. 
im
f (i  1)  f (i )   f (n  1)  f (m)
im

Other useful formulas that are useful in this module:

n n
1.  c   cn
i 1 i 1
n
n  n  1
2. i 
i 1 2
n
n  n  1 (2n  1)
3.  i 2 
i 1 6
 n(n  1) 
n 2

4.  i   3

i 1  2 

Illustrative Examples.

1. Evaluate the following using the properties of sigma notation.

12 12 5 8
a.  3 b.  3 c. i d.  2i  3
i1 i 4 i 1 i 1
Solutions:
n n
a. Since i =1 you can use  c   cn
i 1 i 1
12

 3 = 3(12) = 36
i1
n n
b. Since the lower bound is not equal to 1, consider  cai  c ai
im i m
12

 3 = 3(12-4 + 1) = 3(9) = 27
i 4

3
 n
n  n  1
c. Consider i 
i 1 2
5

i=
5(5+1) 30
= = 15
2 2
i 1
n n n

d. Consider the property  a  b   a   b and other properties

i m
i i
i m
i
i m
i

that applies.
8 8 8

 2i  3 = 2  i   3 = 2 [
8(8+1)
] − 3(8) = 72 – 24 = 48
2
i 3 i 1 i 1

2020
1
2. Evaluate ii 1
2
 3i  2
Solution:
1 a b
Rewrite in the form  such that a (n  2)  b(n  1)  1
i  3i  2 2
n 1 n  2
2020 2020
1 1 1
Then,  2 =  
i 1 i  3i  2 i 1 i  1 i2
It follows that the series
1 1 1 1 1 1 1 1 1 1
=     .   .....    
2 3 3 4 4 5 2020 2021 2021 2022
1 1 505
=  
2 2022 1011
10
3. Evaluate  i(i  5)
i 1
Solution:
10 10 10(10  1)  2(10)  1 10(10  1) 
 i(i  5) =  i
i 1 i 1
2
 5i =
6
5
 2 
2310  110 
=  5   385  275  110
6  2 
ACTIVITY 1

What’s my Index and my Terms?

Direction. For each of the following, state the index and the number of terms
Sigma Notation index of summation # of terms
6
1.  5i
i 1
______ ______

8
2. k
k 3
2
k ______ ______

4
 6
3.  t (t  2)
t 0
______ ______

4
4. i
i 1
2
______ ______

6
5. n
n 5
2
n3 ______ ______

ACTIVITY 2.
Expand and Add

Direction. Write each expression in Expanded form and find the sum
6
1.  (5i  1)
i 1
______________________________ _____

6
2. 2
k 0
k
______________________________ _____

4
3.  (1) t
t 1
t
______________________________ _____

r
5
1
4.  32   ______________________________ _____
r 1 2

3
(1)k 1
5. 
k 0 k  1
______________________________ _____

ACTIVITY 3
Express to Impress

Use sigma notation to express each of the following:

1. 5 + 9 + 13 + 17+…..+ 45 __________

2. -1 + 2 -3 + 4 – 5 + 6 - ….+ 20 __________

3. 31  32  33  34 __________

5
 1 1 1 1
4. 1     ....  __________
2 3 4 100

1 1 1 1 1
5.     __________
4 7 10 13 16

ACTIVITY 4.
Fun for Sum
Direction. Evaluate the following sums using properties of sigma notation.

20
1.  (3i  4)
i 1
__________

5
2. k
k 0
2
 3k  1 __________

4
t 1
3. t2
t 1
__________

20
4.  12
i12
__________

5
5.  (2r  3)
r 1
2
__________

6
ENRICHMENT ACTIVITY
Fun with Telescoping Sum
Task 1: Fun with Telescoping Sum

Challenge
Use
telescoping
sum to
evaluate
the given
sigma
notation.

Task 2: Prove it!