# Arithmetic Series: Sum of the first n terms of an arithmetic sequence

If we ask a student to add numbers from 1 to 1000, for example, how long will it take him to complete this operation?

Perhaps it would be easy if we asked him to add numbers from 1 to 10. But when a series of operations is long, accomplishing them becomes arduous and requires time and effort.

Can math shorten this task? In other words, is there a mathematical formula that directly calculates the sum from 1 to 1000?

Note with me carefully that the expression:

It is an arithmetic sequence whose first term is 1 and its last term is 1000 and the common difference is 1. (The points mean that the sequence continues in the same pattern).

The goal of this lesson is to find how to add these terms in this way:

This is what we will try to answer in this section of the Arithmetic Series lesson.

## Arithmetic Sequence vs Arithmetic Series

The difference between Arithmetic Sequence and Arithmetic Series is:

### Note:

• An arithmetic sequence is a set of terms in which there is a relation between the term and the term before or after it
• In the expression, we separate these terms with a comma:
• Terms of an arithmetic sequence are arranged (either increasing or decreasing)
• An arithmetic series is the sum of the terms of an arithmetic series:

The sum of the first n terms of this arithmetic sequence denoted by and it is called the partial sum and we write

Let's take the following example:

Let be an arithmetic sequence where:

Then,

But how do we find ? Adding terms in this way, as we said before, requires a lot of time and effort. So we need a mathematical formula that sums it all up.

we know that:

We rewrite this equation, but this time we start descending from the last term to the first term.

Adding the equations (1) and (2) side by side:

### Note:

Let an arithmetic sequence.

The sum of n first terms of the sequence is:

• : first term
• : nth term
• : number of terms to add

The expression is written in terms of the general term :

We substitute into the formula:

### Some illustrative examples of arithmetic series.

#### Example1:

Let's go back to the example we started from in this lesson. What is the sum of the numbers from 1 to 1000?

#### Solution:

This example is very easy, then we will be able to calculate more complex sums.

The numbers from 1 to 1000 are an arithmetic sequence:

• Its first term is:
• Its last term is:
• The number of terms is:

therefore,

#### Example2:

Find the sum of the terms of the following arithmetic series:

#### Solution:

This example differs from the previous example in that the number of terms in this example is unknown.

To calculate n we return to the formula that enables us to calculate the number of terms in an arithmetic sequence:

• is the nth term. ()
• is the first term. ()
• is the common difference. ()

therefore:

So the partial sum of is:

#### Example3:

Find the sum of the first 69 terms of the following arithmetic sequence:

#### Solution:

Always in the same way, we add the terms using the following formula:

• is the first term. ()
• is the numbers of terms. ()
• is the 69th term. (we must calculate it using the term general formula)

Here we need to know the common difference:

Therefore,

Thus,

#### Example4:

let an arithmetic sequence in which and

Calculate .

#### Solution:

In this example, we only know the 5th term and the 13th term.

Let's calculate the common difference from the general term formula:

Therefor,

But we don't know and . Let's calculate it.

And,

Therefor,

#### Example5:

If the sum of terms of an arithmetic sequence, How many terms are there in this sequence?

#### Solution:

In this example we know the sum of the terms , but we do not know the number of these terms

Let's turn the problem into an algebraic expression:

The partial sum formula is:

So, the arithmetic sequence has 28 terms (Now we can calculate the common difference).