*When the terms of an arithmetic sequence are added together, an arithmetic series is formed.*

Suppose a sequence of numbers is arithmetic and you want to find the sum of the first *n* terms. There are two different equations that can be used.

**(1)**

Where:

refers to the sum of the first terms in a series

refers to the number of terms in an arithmetic series

refers to the first term in the series

refers to the last term

OR

**(2)**

Where:

refers to the sum of the first terms in a series

refers to the number of terms in an arithmetic series

refers to the first term in the series

refers to the common difference

So which equation do you use?

It depends on the information provided in the question.

If you have or need to calculate the value of then you will use the first equation.

If you have or need to calculate the value of then you will use the second equation.

**The first term of a sequence is 14 and the sum of the first 15 terms is 975.
**

The information in the question tells us that:

the 15th term

Given this information, we must use the first formula to calculate what the 15th term is.

where

The 15th term is 116.

**The first term of an arithmetic sequence is -5 and the sixth term is 30.
**

From the question we know:

We now need to find what the value of is:

Now that we know what is, we can use the formula to answer the question.

The values that we have are:

We will therefore use the second formula.

Where and , the sum of the first ten terms in 265.

**Find the sum of the first six given terms of an arithmetic sequence**

** **

The information in the question tells us that:

Given this, we could use either formula as we have all of the information for both.

OR

As you get the same result for both formulas, it is up to you which to use!

Arithmetic Sequence

Finding the Terms of an Arithmetic Sequence

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