# Sequences, Series and Binomial Theorem

## Arithmetic Sequences


### Determine if a Sequence is Arithmetic

The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence,  is d, the common difference, for n greater than or equal to two.
In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.
If we know the first term,  and the common difference, d, we can list a finite number of terms of the sequence.

### Find the General Term (nth Term) of an Arithmetic Sequence

Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is  and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with .
The first term adds 0d to the , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to  is one less than the number of the term. This leads us to the following
We will use this formula in the next example to find the 15th term of a sequence.
Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.
Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.

### Find the Sum of the First n Terms of an Arithmetic Sequence

As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum,  of the first  terms of any arithmetic sequence is written as  To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.
We can develop this new formula by first writing the sum by starting with the first term,  and keep adding a d to get the next term as:
We can also reverse the order of the terms and write the sum by starting with  and keep subtracting d to get the next term as
If we add these two expressions for the sum of the first n terms of an arithmetic sequence, we can derive a formula for the sum of the first n terms of any arithmetic series.
Because there are n sums of  on the right side of the equation, we rewrite the right side as 
We divide by two to solve for 
This gives us a general formula for the sum of the first n terms of an arithmetic sequence.
We apply this formula in the next example where the first few terms of the sequence are given.
In the next example, we are given the general term for the sequence and are asked to find the sum of the first 50 terms.
In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first n terms.

### Key Concepts

1. General Term (
The general term of an arithmetic sequence with first term  and the common difference d is

2. Sum of the First 
The sum,  of the first n terms of an arithmetic sequence, where  is the first term and  is the nth term is


### Practice Makes Perfect

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 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 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 50 terms of the arithmetic sequence whose general term is given.
In the following exercises, find each sum.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?