The document discusses harmonic and Fibonacci sequences. A harmonic sequence is one whose reciprocals of terms form an arithmetic sequence. Examples of harmonic sequences are provided. The nth term of a harmonic sequence is given. It is also explained that a Fibonacci sequence is one where the first two terms are 1, and each subsequent term is the sum of the previous two. The nth term of a Fibonacci sequence cannot be determined by a single formula. Examples of Fibonacci and Fibonacci-like sequences are provided.

1. Harmonic and Fibonacci
Sequence

2. Review!
What is an arithmetic sequence?
Give examples of an arithmetic sequence.

3. Harmonic Sequence
A sequence whose reciprocals of terms form an
arithmetic sequence.

4. Examples!
1
2
,
1
4
,
1
6
,
1
8
, …
−
2
3
, −
2
7
, −
2
11
, −
2
15
, …
5
7
,
1
2
,
5
13
,
5
16
, …
4
9
,
1
3
,
4
15
,
2
9
, …

5. nth term of a Harmonic Sequence
𝑎𝑛 =
𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟
𝑎1 + 𝑛 − 1 𝑑
Find the 10th term!
1.
2
3
,
2
7
,
2
11
,
2
15
, …
2.
5
7
,
1
2
,
5
13
,
5
16
, …

6. Determine whether the given sequence is
harmonic or not then solve for the 8th term.
1.
1
5
,
1
10
,
1
15
,
1
20
, …
2.
3
8
,
1
4
,
3
16
,
3
20
, …
3.
7
9
,
7
15
,
1
3
,
7
26
, …

7. How do you describe harmony?
1
5
,
1
10
,
1
15
,
1
20
, …
RESPECT for HUMAN DIGNITY.
Our individual differences(different denominators)
will lead to harmonious relationship for as long as
we respect each other.

8. Will you change your personality for a
harmonious relationship?
7
9
,
7
15
,
1
3
,
7
26
, …
We are all different, and these differences should
not hinder us in attaining harmony. Change if you
can change and if there should be changed.

9. Romans 14:19
Let us therefore follow after the things which
make for peace, and things wherewith one may
edify another

10. Fibonacci Sequence
A sequence whose 𝑎1 = 𝑎2 = 1 and the next terms
are obtained by adding the two preceding terms.
1,1,2,3,5,8, …
There is only one Fibonacci Sequence

11. Fibonacci – Like Sequence
Sequence that follow the same pattern as the
original Fibonacci Sequence.
2,2,4,6,10, …
3,3,6,9,15, …
10,10,20,30,50, …

12. nth term of a Fibonacci Sequence
There is no other way in finding the nth term of a
Fibonacci or Fibonacci – like sequence than listing
and adding the preceding terms manually.

13. Determine whether the following
statements are TRUE or FALSE.
1. The 3rd term of a Fibonacci or any Fibonacci –
like sequence is always twice the first term.
2. The 5th term of a Fibonacci or any Fibonacci –
like sequence is divisible by 5.
3. The 6th term of a Fibonacci – like sequence with
1st term 4 is 32.
4. The sum of the first 5 terms of a Fibonacci
sequence is 12.

14. Drill
Determine whether the following is harmonic or
not then find 𝑎9.
1.
1
7
,
1
13
,
1
19
,
1
25
, …
2.
4
19
,
1
5
,
4
21
,
2
11
, …
3.
6
5
,
2
3
,
6
13
,
6
17
, …

15. Drill
Find the 8th term of the following Fibonacci – like
sequence if:
1. The 2nd term is 5.
2. The 3rd term is 8.
3. The 1st term is 7.