pre=calculus topic pascal triangle and binomial theorem

1. Pascal’s Triangle and the
Binomial Theorem

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

3. 1. Determine the binomial
coefficients using Pascal’s Triangle

4. 2. Derive the binomial
Theorem;

5. 3. apply the binomial Theorem in
expanding a binomial

6. Can you recall the formula for the
square of a binomial? How about
the cube of a binomial? Their
respective formulas are as
follows:
=
=

9. (𝑥+𝑎)
𝑛
=∑
𝑘=0
𝑛
(𝑛
𝑘)𝑥
𝑘
𝑎
𝑛−𝑘

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,