The document discusses harmonic sequences, arithmetic sequences, and Fibonacci sequences. A harmonic sequence is one where the reciprocals of the terms form an arithmetic sequence. An arithmetic sequence is one where the differences between consecutive terms are constant. A Fibonacci sequence is one where each term is the sum of the two preceding terms, starting with either 1,1 or 0,1 as the first two terms. The document provides examples of finding terms in harmonic sequences given initial terms and differences, and examples of extending Fibonacci sequences by calculating subsequent terms.

1. Harmonic Sequence

2. A harmonic sequence is a sequence such
that the reciprocals of the terms form an
arithmetic sequence.
The sequence is a harmonic sequence
since their reciprocals 3, 6, 9, 12 form an arithmetic
sequence.

3. Arithmetic Sequence Harmonic Sequence
2, 4, 6, 8,…
18, 12, 6, 0, -6, -
12

4. Illustrative Examples
A. Find the first six terms of a harmonic sequence where
is 1/3 and d is 1/5
.
B. Find the 13th term of a harmonic sequence if = 1/6
and d = 1/2
C. Given the arithmetic sequence -20, -26, -32, -38,…,
find the first 8 terms of the
corresponding harmonic sequence.
D. Given the arithmetic sequence ½, 1, 3/2, 2,…,
find the 10th term of the corresponding harmonic sequence.

5. Fibonacci Sequence

6. A Fibonacci sequence is a sequence where its first
two terms are either both 1, or 0 and 1; and each term,
thereafter, is obtained by adding the two preceding terms.
Illustrative Examples
A. What is the next 5 terms in the Fibonacci sequence
0, 1, 1, 2, 3, 5,…?
B. Given the Fibonacci sequence 5, 8, 13, 21, 34,…,
find the next 6 terms.