# Given and

In the applet below, you are given the first term, and the common difference. You are then asked to calculate one of the other terms of the sequence.

Notice the calculation that you are repeating each time. Let’s call the first term and the common difference . What calculation do you perform each time you wish to calculate

1. ?
2. ?
3. ?

The answer to the third question is the a formula that may be used to calculate the general term for an arithmetic sequence.

## Example 1

An arithmetic sequence has first term and common difference . Find an expression to calculate any term of the sequence, .

Solution

The first five terms of the sequence are

We know that . We can replace with and we can replace with .

It’s helpful to put the 3 before the brackets:

We can conclude that .

Now we can use this formula to find any term in the sequence: let’s find the fifth term. We replace the value with .

.

We can confirm this calculation with the sequence written out above.

## Example 2

An arithmetic sequence has first term and common difference . Find an expression to calculate any term of the sequence, .

Solution

The first five terms of the sequence are

We know that . We can replace with and we can replace with .

It’s helpful to put the (-3) before the brackets, remember to take the negative sign with you:

We can conclude that .

Now we can use this formula to find any term in the sequence: let’s find the fourth term. We replace the value with .

.

We can verify this calculation with the sequence written out above.

Example 3

A regular non-agon has the first angle positioned at 10 degrees, second angle at 50 degrees, etc.

(a) Write out the sequence of angles for all nine vertices.

(b) Calculate the nth term for .

(c) Use the formula to verify that the last angle (vertex number 9) is at .

Solution:

(a) The angles are positioned at

(b) The common difference is .

(c) The formula says that where in this case, represents the number of the vertex.

At the ninth vertex, . The formula says as required.

# Practice

Explore the Arithmetic Sequences problem sets on Transum Math.

Or, find the general term of the problems given here: