pre-calculus topic Sequence and Series

1. Sequence and Series

2. At the end of the lesson, the student is
able to

3. 1. illustrate sequence
and series

4. 2. deferentiate a series
from a sequence

5. 3. use the sigma notation to
represent a series

6. Sequence: A Review
In your previous
mathematics subjects,
you learned about the
succession of numbers
that follows a pattern.
For example, 1, 3,5 and
7, what do you think
will be the next?

7. It is 9. We can notice
from the given that
the next term is
obtained by adding 2
to the previous term.
This succession of
numbers is what we
call a sequence. A
sequence is either
finite or infinite.

8. Definition:
A Sequence is a succession of numbers that
follows a pattern. An infinite sequence is a
function whose domain is the set of
consecutive positive integers. If the domain
is the set of the first n positive integers,
then it is said to be a finite sequence. The
individual elements in a sequence are
called terms.

9. The first term of the sequence is
denoted as the second as and so on
with , or the nth term of the
sequence, as the value of the
sequence at .

10. Example1.
Give the first five terms of the sequence represented by the
formula by
Example 2: Give the first three terms of the sequence defined by

11. Arithmetic Sequence
Definition:
An arithmetic sequence where the difference between two
consecutive terms is constant, that is, each term is obtained by
adding a nonzero constant, called common difference, to its
preceding term of an arithmetic sequence is given by
Where - is the th term
- is the first term; and
- is the common difference.

12. 4.Example: Consider the arithmetic
sequence given by Find the first five
terms of the sequence.
5. Example: Given the arithmetic
sequence 7,13, 19, 25,31,……determine
the 20th
and 21st
terms.

13. 6. Example: If the first term is 2 and the
third term is 12, find the sixth term of this
arithmetic sequence.

14. Geometric Sequence
A geometric sequence is a sequence where the second and
succeeding terms are obtained by multiplying the preceding
term by a nonzero constant called common ratio. The nth term
of a geometric sequence is given by
Where - is the nth term
- is the first term
- is the common ratio.

15. 7.Example: Find the first four terms of the geometric sequence defined
by
8. Example: Given the geometric sequence 2, -6, 18, -54, 162……….,
determine the 10th
and 11th
terms.
9. If the first and the fourth terms of a geometric sequence are 2 and ,
respectively. Find the common ratio of the sequence.

16. Review of Summation:

21. Series
Now that we have recalled how to find the nth term of a sequence, we
then recall how to determine the sum of a specified number of terms.
This sum is referred to as series. Let us see the next example.

22. Example 10. Given the sequence 1,3,5,7,9,11,13,…, the sum of the five
terms is given by , , , , = 1+3+5+7+9 = 25.
Another way of writing this sum is by using the Greek capital letter
sigma denoted as . Thus, the above equation can be written as
= , , , , = 1+3+5+7+9 = 25.

23. Arithmetic Series
The Sum of a finite sequence is given by
.
In sigma notation, we write the formula as
= + = .

24. Example 11. Using the previous formula, we can now determine the
sum of the first 1,000 positive integers, that is,