The document discusses various types of sequences, focusing on the Fibonacci sequence and harmonic sequences. It describes the Fibonacci sequence's definition and properties, including its generation based on preceding terms. Additionally, it outlines how harmonic sequences are formed from the reciprocals of terms in an arithmetic sequence.

1. Other Sequences
( Fibonacci, Harmonic,…)
MARK O. AGUSTIN, LPT
MATHEMATICS TEACHER

2. OBJECTIVES
• Illustrate other types of sequences;
• Solve problems involving
sequences.

3. Fibonacci Sequence

4. FIBONACCI
• In 1202, Leonardo of
Pisa, who was called
Fibonacci, wrote a book
on arithmetic and
algebra titled Liber
Abaci.
• The sequence 1 1 2 3
5 8 13 … LEONARDO OF PISA

5. FIBONACCI
SEQUENCE
• The sequence 1 1 2
3 5 8 13 …
Is called the Fibonacci
Sequence. The first two
terms of this sequence
are 1 and each
successive term is the
sum of the preceding
terms.
Add more information here as needed

6. GENERAL TERM OF
FIBONACCI- TYPE SEQUENCE
• A Fibonacci type
sequence is a sequence
in which the general term
is given by the formula
Where and are given.
The Fibonacci sequence
corresponds to = =1
GOLDEN RATIOS

7. EXAMPLE
1. If and represent any first numbers, list the
first eight terms of this Fibonacci type
sequence.
2. Given and . Find .
3. Find the 30th
term of the Fibonacci
sequence

8. Harmonic Sequence

9. Harmonic
Sequence
• If are terms of an
arithmetic sequence,
then the sequence of
the reciprocals of these
terms, , is called a
harmonic sequence.
FIRST MEMBER OF HARMONIC
SEQUENCE

10. EXAMPLE
1. The sixth term of the harmonic sequence .
2. The first term of a harmonic sequence whose
fourth term is and whose twelfth term is