Series.pptxvvggghgufifudududydydydudyxyxyx

Module 6: Series
Marvin C. Santos
Special Science Teacher I

In this module, you will be able to
• Illustrate series
• Differentiate series from a sequence
• Represent a series using sigma notation
• Expand a series given its equivalent sigma notation
• Solve problems involving arithmetic and geometric series
• Solve real life problems involving series

Math Challenge
What is the missing number
2,5,8, __,14
0,1,3, __,15,31
11
7
80
0,2,8,26, __

SERIES
Basically a Series is the sum of the finite or infinite number if terms in a
sequence.
Let the terms of an infinite sequence be
A finite series is the sum of the first n terms of a sequence.it is also reffered to
as the nth partial sum of a sequence. A finite series is denoted by
𝑠𝑛=∑
𝑖=1
𝑛
𝑎𝑖=𝑎1+𝑎2+𝑎3+...+𝑎𝑛
𝑎1,𝑎2 ,...,𝑎𝑗 ,...

SERIES
An infinite series is the sum of all terms of an infinite sequence. It is denoted
by
The Greek capital letter sigma is used to indicate summation.
Other names used for this symbol are series notation, summation notation, and
sigma notation.
𝑠𝑛=∑
𝑖=1
∞
𝑎𝑖=𝑎1+𝑎2+𝑎3+...+𝑎𝑖+...
∑❑

Write the terms and evaluate the series
 
5
1
2
i
i



 
5
1
2 (1 2) (2 2) (3 2) (4 2) (5 2)
i
i

          

 
5
1
2 3 4 5 6 7
i
i

     

 
5
1
2 25
i
i

 


Write and evaluate the series in each notation
6
1
2
3i
i


  
4
1
2
n
n
n


   
8
3
1 1
i
n n

 

3 267 193
3 894

Write the following series using sigma notation
• 3 + 6 + 9 + 12 + 15 + 18 + 21
• 2 + 5 + 8 +11
• 2 + 5 + 10 + 17 +26
• -8 – 18 – 32 – 50 – 72 + …

Write and evaluate the following series
• Write and evaluate the series in each number. AYOKO NG DECIMAL
NUMBER
5
0
2
2
n
n
n




11
3
3 2
1
n
n
n




 
5
1
1
n
n
n



 
15
1
8 4
k
k



 
7
2
1
10
k
k



   
13 8
1 1
10 4 10 4
i i
i i
 
  
 
 
15 5
2 2
1 1
4 3
k k
k k
 
 
 

Write and evaluate the following series
• Write and evaluate the series in each number.
5
0
2
2
n
n
n




11
3
3 2
1
n
n
n




 
5
1
1
n
n
n



 
15
1
8 4
k
k



 
7
2
1
10
k
k



   
13 8
1 1
10 4 10 4
i i
i i
 
  
 
 
15 5
2 2
1 1
4 3
k k
k k
 
 
 
-13/35
-64591/5544
-3
900
-70
1 186
1 475

Write each series using sigma notation
5 7 9 11 13 15 17
     
1 1 1 1
...
4 8 16 32
   
9 16 25
...
4 5 6
  

Ultimate Item
37 65 101
13 25 41
2 2 2
    

Given the infinite series
Find the following
1. Partial sum of the first four terms in the series
2. Sum of the series
1
5
10n
n




Given the infinite series
Determine the following:
• Third partial sum of the series
• Fifth partial sum of the series
• Eight partial sum of the series
• Tenth partial sum of the series
• Sum of the series
1
1
2n
n




Find the sum of the series as indicated
 
6
2 2
3
1
n
n n

 
 
 

 
5
1
!
2 ! 1
k
k
k
 

 
6
2
1
2 1
n
n n




  
9
0
2 3
k
k k

 

 
4
0
1
1 2
k
k
k




Find the sum of the series as indicated
 
4
0
1
1 2
n
n
k



 
6
2 2
3
1
n
n n

 
 
 

 
5
1
!
2 ! 1
k
k
k
 

 
6
2
1
2 1
n
n n




  
9
0
2 3
k
k k

 

40
1195417/370689 120
6949/45045
481/1530

In this part of the module, you will be able to
• Review sequence
• Illustrate arithmetic series
• Solve problems involving arithmetic series

Review
Write the following in sigma notation
1. 5 + 7 + 9 + 11 + 13 + 15 + ….
2. 1+ + + + + …
3. -3 – 2 – 1 + 0 + 1 + 2 + 3 + ….

Lets play a game!
Identify what are the words behind the
jumbled letters

ARITHMETIC SERIES
Marvin C. Santos

Recall arithmetic sequence
Arithmetic Sequence
• In an Arithmetic Sequence the difference between one term and the next
is a constant.
Example
1 , 4 , 7 , 10 , 13 , 16 , 19 , 21

Arithmetic
Sequence Rule
 
1 1
n
a a n d
  
1
n
a last term
a first term
n number of terms
d common difference





Arithmetic Series
The sum of the first n terms of an arithmetic sequence is referred to as arithmetic series.
The sum of an arithmetic sequence, denoted by
Is given by the formula
Where n is the number of terms
𝑆𝑛
 
1
2
n n
n
S a a
 

Extend your knowledge
It is sometimes necessary to express the sum of the first n terms of an
arithmetic sequence in terms of and the common difference d.
Thus, the sum of the first n terms of an arithmetic sequence can also be
computed using the formula
1
a
 
1
2 1
2
n
n
S a n d
  
 
 

Sum of an Arithmetic Sequence
 
1
2
n n
n
S a a
 
 
1
2 1
2
n
n
S a n d
  
 
 

Example 1
Find the sum of the first 35 terms of an arithmetic sequence whose first term
is 3 and the common difference is 6
Observe that the given values:
1 3
6
35
a
d
n



 
35
35
2(3) 35 1 6
2
S   
 
 
 
35
35
6 204
2
S  
35 3675
S 
 
1
2 1
2
n
n
S a n d
  
 
 

Example 2
Find the sum of the series
4 +9 + 14 + 24 + … + 189

Read analyze the following problems. Write your answer on a sheet of
paper
• Find the sum of the first 50 terms of an arithmetic sequence whose first
term is 2 and the common difference is 4
• Find of the series

What have I learned so far?
• Find the sum of the series
• Find the sum of the series
3 11 19 ... 227
   
3 3 5 3 7 3 ... 39 3
   

Group Activity
• Each group will be given sets of problem sets and they are required to find
the sum of each arithmetic series
• The teacher will give the sets of activity/work sheets

Find the sum if the series
3 + 11 + 19 + … + 227
Find the sum of the series
3 3 5 3 7 3 ... 39 3
   
Find 12
S of the sequence
7 3
n
a n
 
Find the sum of the first
40 consecutive integers
starting from 23
Find the sum of the first
20 odd integers starting
from 1
12
S

Series.pptxvvggghgufifudududydydydudyxyxyx

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