# Sequences, Series and Binomial Theorem

## Binomial Theorem


### Use Pascal’s Triangle to Expand a Binomial

In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded 
To expand  we recognize that this is  and multiply.
To find a method that is less tedious that will work for higher expansions like  we again look for patterns in some expansions.
Notice the first and last terms show only one variable. Recall that  so we could rewrite the first and last terms to include both variables. For example, we could expand  to show each term with both variables.
Generally, we don’t show the zero exponents, just as we usually write x rather than 1x.
Let’s look at an example to highlight the last three patterns.
From the patterns we identified, we see the variables in the expansion of  would be
To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.
The array to the right is called Pascal’s Triangle. Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
This triangle gives the coefficients of the terms when we expand binomials.
In the next example, we will use this triangle and the patterns we recognized to expand the binomial.
In the next example we want to expand a binomial with one variable and one constant. We need to identify the a and b to carefully apply the pattern.
In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the a and b of the pattern, we must once again carefully apply the pattern.

### Evaluate a Binomial Coefficient

While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.
To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation  which is called a binomial coefficient. We read  as “n choose r” or “n taken r at a time”.
In the previous example, parts (a), (b), (c) demonstrate some special properties of binomial coefficients.

### Use the Binomial Theorem to Expand a Binomial

We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.
Notice that when we expanded  in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle.
The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.
Things can get messy when both terms have a coefficient and a variable.
The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem.
Notice, that in each case the exponent on the b is one less than the number of the term. The  term is the term where the exponent of b is r. So we can use the format of the  term to find the value of a specific term.

### Key Concepts

1. Patterns in the expansion of 


2. Pascal’s Triangle


3. Binomial Coefficient 
: A binomial coefficient  where r and n are integers with  is defined as

We read  as “n choose r” or “n taken r at a time”.
4. Properties of Binomial Coefficients

5. Binomial Theorem: For any real numbers a, b, and positive integer n,


### Section Exercises


### Practice Makes Perfect

Use Pascal’s Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
Evaluate a Binomial Coefficient
In the following exercises, evaluate.
Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial.
In the following exercises, find the indicated term in the expansion of the binomial.
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises


### Sequences

Write the First Few Terms of a Sequence
In the following exercises, write the first five terms of the sequence whose general term is given.
Find a Formula for the General Term (
In the following exercises, find a general term for the sequence whose first five terms are shown.
Use Factorial Notation
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
Find the Partial Sum
In the following exercises, expand the partial sum and find its value.
Use Summation Notation to write a Sum
In the following exercises, write each sum using summation notation.

### Arithmetic Sequences

Determine if a Sequence is Arithmetic
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.
Find the General Term (
In the following exercises, find the term described using the information provided.
In the following exercises, find the indicated term and give the formula for the general term.
In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.
Find the Sum of the First 
In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.
In the following exercises, find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.
In the following exercises, find each sum.

### Geometric Sequences and Series

Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
Find the General Term (
In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.
In the following exercises, find the indicated term of the given sequence. Find the general term of the sequence.
Find the Sum of the First 
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
In the following exercises, find the sum
Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
In the following exercises, write each repeating decimal as a fraction.
Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.

### Binomial Theorem

Use Pascal’s Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
Evaluate a Binomial Coefficient
In the following exercises, evaluate.
Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial, using the Binomial Theorem.
In the following exercises, find the indicated term in the expansion of the binomial.
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

### Practice Test

In the following exercises, write the first five terms of the sequence whose general term is given.
In the following exercises, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.
In the following exercises, find the sum.