HARMONIC PROGRESSION

1
HARMONIC PROGRESSION
2
What is a Harmonic Progression?
3
A Harmonic Progression is a sequence of
quantities whose reciprocals form an arithmetic
progression.
4
Note!
5
The series formed by the reciprocals of the
terms of a geometric series are also geometric
series.
6
And
7
There is no general method of finding the sum
of a harmonic progression. 
8
Example
9
The Sequences1 , s2 , , snis a Harmonic
Progression if 1/s1 , 1/s2 , , 1/snforms
an Arithmetic Progression.

                                                                                
 
10
Method For Re-checking a Harmonic Progression
11
A Harmonic Progression is a set of values that,
once reciprocated, results to an Arithmetic
Progression. To check , the reciprocated values
must possess a rational common difference. Once
this has been identified, we may say that the
sequence is a Harmonic Progression.
12
Harmonic Means are the terms found in between two
terms of a harmonic progression.
13
Problems
14
Determine which of the following are Harmonic
Progressions.
15
1 ,1/2 , 1/3 , 1/4 , ...
16
Step 1 Reciprocate all the given terms. The
reciprocals are 1 , 2 , 3 , 4 , Step 2
Identify whether the reciprocated sequence is an
Arithmetic Progression by checking if a common
difference exists in the terms.
17
Answer It is a Harmonic Progression.
18
2) 1 , 1/4 , 1/5 , 1/7 , ...
19
Step 1 Reciprocate all the given terms. The
reciprocals are 1 , 4 , 5 , 7 , Step 2
Identify whether the reciprocated sequence is an
Arithmetic Progression by checking if a common
difference exists in the terms.
20
Answer It is NOT a Harmonic Progression.
21
Determine the next three terms of each of the
following Harmonic Progressions.
22
1) 24 , 12 , 8 , 6 ,
23
Solution 24 , 12 , 8 , 6 , 1/24 , 1/12 ,
1/8 , 1/6 To find the common difference
1/12 1/24 2/24 1/24 1/24
24
Note!
25
You can subtract the second term to the first
term, the third to the second term, the forth to
the third term, and so on and so forth.
26
To get the next three terms 5th Term 1/6
1/24 4/24 1/24 5/24
Reciprocate 24/5
27
6th Term 5/24 1/24 6/24 1/4
Reciprocate 4
28
7th Term 1/4 1/24 6/24 1/24
7/24 Reciprocate 24/7
29
Find the Harmonic Mean between the following
terms.
30
1) 12 and 8
31
Step 1 Reciprocate all the given terms. The
reciprocals are 1/12 and 1/8Step 2 Arrange
the given terms as follows
32
1/12 Harmonic Mean 1/8 1st term
2nd term 3rd term
33
For this problem, we will use the formulatn
t1 (n 1)d
34
We may now substitute the values in the problem
to the formula to find the common difference (d)
and the Harmonic Mean as follows
35
t3 t1 (3 - 1)d1/8 1/12 2d1/8 1/12
2d(3 2) / 24 2d(3 2) 48d1 48dd
1/48
36
After getting the Common Difference, add it to
the first term to get the Harmonic Mean between
the two terms.
37
t2 t1 d 1/12 1/48 (4 1) / 48
5/48Reciprocate 48/5
38
Insert three Harmonic Means between the following
terms
39
1) 36 and 36/5
40
Step 1 Reciprocate all the given terms. The
reciprocals are 1/36 and 5/36Step 2 Arrange
the given terms as follows
41
1/361st termHarmonic Means2nd , 3rd ,
and 4th term5/365th term
42
For this problem, we will use the formulatn
t1 (n 1)d
43
We may now substitute the values in the problem
to the formula to find the common difference (d)
and the Harmonic Means as follows
44
t5 t1 (5 - 1)d5/36 1/36 4d5/36 1/36
4d(5 - 1) / 36 4d(5 - 1) 144d4 144dd
4/144 1/36
45
After getting the Common Difference, add it to
the first term, then add it to the second term,
and then add it to the third term to get the
Harmonic Means between the two terms.
46
t2 t1 d 1/36 1/36 2/36
1/18Reciprocate 18
47
t3 t2 d 2/36 1/36 3/36
1/12Reciprocate 12
48
t4 t3 d 3/36 1/36 4/36
1/9Reciprocate 9
49
Therefore, the three means between 36 and 36/5
are 18, 12, and 9.
50
Activity
51
Determine if the following are harmonic
progressions or not1) 1/12 , 1/24 , 1/362) 2
, 5 , 7 , 83)1/5 , 1/10 , 1/15
52
Find the next three terms in the following
harmonic progressions1) 1/2 , 1/5 , 1/8 , 1/11
, 2) 19 , 17 , 15 , 13, 3) 12 , 6 , 4 , 3 ,
53
Find the harmonic mean between1) 1/2 and
1/52) 1 and 1/9
54
Insert three harmonic means between1) 1/2 and
1/82) 1 and 1/10
55
Homework
56
JoKe LanG!aKaLa nYo hA!(CorNy dB?)
57
Prepared byTHE THREE MUSKETEERSa.k.a.Lucas
FerrerPaul Steven SantosandEarl Jeremy Buera