The document discusses harmonic sequences. A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence. It provides examples of determining terms of harmonic sequences. It explains that the terms between any two terms of a harmonic sequence are called harmonic means. An example is worked out of inserting two harmonic means between two given terms of a harmonic sequence.

1. HARMONIC
SEQUENCE
Ms. G. Martin

2. Let’s Learn!
Study the following sequences.
1. 1,
1
5
,
1
9
,
1
13
,
1
17
, …
2. 1,
1
2
,
1
3
,
1
4
, …
3. 1,
1
4
,
1
7
,
1
10
, …
Which sequence are geometric? Which sequence are arithmetic?

3. If we take the reciprocal of each term of the sequences, we have
1, 5, 9, 13, 17, …
1, 2, 3, 4, …
1, 4, 7, 10, …
What kind of sequences are formed? How do we define a
harmonic sequence?
A harmonic sequence is a sequence whose reciprocals of
the terms form an arithmetic sequence.

4. Example 1.
Determine the 101st term of the harmonic sequence 1,
1
5
,
1
9
,
1
13
,
1
17
, …
Steps Solution
1. Determine the reciprocal of
each term of 1,
1
5
,
1
9
,
1
13
,
1
17
,
…
1, 5, 9, 13, 17, …
The result is an arithmetic
sequence.
2. Find the common difference of
the arithmetic sequence.
d = 5 – 1 = 4

5. Steps Solution
3. Solve for the 101st term of the
arithmetic sequence 1, 5, 9, 13,
17, …
If d = 4 and A1 = 1, we have
An = A1 + (n – 1)d
A101 = A1 + ( 101- 1)d
= 1 + (101 – 1)4
= 1 + (100)4
= 1 + 400
A101 = 401
4. Take the reciprocal of the
101st term obtained in step 3.
A101 = 401 =
𝟏
𝟒𝟎𝟏
So, the 101st term of the harmonic sequence is
𝟏
𝟒𝟎𝟏
.

6. Example 2.
Find the 15th term of the harmonic sequence
1
4
,
1
8
,
1
12
, …
Steps Solution
1. Determine the reciprocal of
each term of
1
4
,
1
8
,
1
12
, …
4, 8, 12, …
The result is an arithmetic
sequence
2. Find the common difference of
the arithmetic sequence.
d = 8 – 4 = 4

7. Steps Solution
3. Solve for the 15th term of the
arithmetic sequence 4, 8, 12, …
If d = 4 and A1 = 4, we have
An = A1 + (n – 1)d
= 4 + (15 – 1)4
= 4 + (14)4
= 4 + 56
An = 60
4. Take the reciprocal of the 15th
term obtained in step 3.
An = 60 =
1
60
So, the 15th term of the harmonic sequence is
1
60

8. Example 3.
Insert two harmonic means between
1
3
and
1
15
.
 The terms between any two terms of a harmonic sequence are
called harmonic means.
Steps Solution
1. Find the reciprocal of each term
1
3
, _____, ___,
1
15
.
3, ___, ___, 15
2. Identify the given terms in the
corresponding arithmetic
sequence.
Here, A1 = 3 and A4 = 15.

9. Steps Solution
3. Find the common difference. If A1 = 3 and A4 = 15, we have
An = A1 + (n – 1)d
A4 = A1 + (4 – 1)d
15 = 3 + 3d
12 = 3d
d = 4
4. Find the two terms between 3 and
15.
If d = 4 and A1 = 3, we have
A2 = A1 + d = 3 + 4 = 7
A3 = A2 + d = 7 + 4 = 11
So, the corresponding arithmetic
sequence is 3, 7, 11, 15.
5. Take the reciprocal of the 2nd term
and 3rd term obtained in step (4).
A2 = 7 =
1
7
A3 = 11 =
1
11

10. Thank You!