# Arithmetic Sequence

How to find the nth term of an arithmetic sequence: definition, formula, 3 examples, and their solutions.

## Definition

### Definition

For the terms of a sequence,

if you add a term and a constant d

then you get the next term.

a_{1} + d = a_{2}

a_{2} + d = a_{3}

a_{3} + d = a_{4}

...

This sequence is an arithmetic sequence.

This d is called the common difference.

It is the difference of the adjacent terms.

(a_{2} - a_{1} = d, a_{3} - a_{2} = d, ...)

## Formula

### Formula

For an arithmetic sequence,

a_{n} = a + (n - 1)d.

a_{n}: nth term

a: First term, a_{1}

d: Common difference

## Example 1

### Example

### Solution

Find the first term a and d.

Write 1, 4, 7, 10, 13.

The first term, a, is 1.

1 + 3 = 4

4 + 3 = 7

7 + 3 = 10

10 + 3 = 13

So write, the d, +3

between the terms.

a = 1

d = 3

So a_{n} = 1 + (n - 1)⋅3.

+(n - 1)⋅3 = +3n - 3

Multiply a Monomial and a Polynomial

1 + 3n - 3 = 3n - 2

So a_{n} = 3n - 2.

## Example 2

### Example

### Solution

Find the a and d.

Write -2, 5, 12, 19.

The first term, a, is -2.

-2 + 7 = 5

5 + 7 = 12

12 + 7 = 19

So write, the d, +7

between the terms.

a = -2

d = 7

So a_{k} = -2 + (k - 1)⋅7.

+(k - 1)⋅7 = +7k - 7

-2 + 7k - 7 = 7k - 9

So a_{k} = 7k - 9.

It says

a_{k} = 551.

And you found

a_{k} = 7k - 9.

So

7k - 9 = 551.

Move -9 to the right side.

Then 7k = 560.

Divide both sides by 7.

Then k = 80.

So k = 80.

## Example 3

### Example

### Solution

To find a_{n},

find the first term a

and the common difference d.

For an arithmetic sequence,

a_{8} = a + 7⋅d.

And it says

a_{8} = 5.

So a_{8} = a + 7d = 5.

For an arithmetic sequence,

a_{12} = a + 11⋅d.

And it says

a_{12} = 13.

So a_{12} = a + 11d = 13.

a + 7d = 5

a + 11d = 13

The goal is to solve this system

and find the values of a and d.

Both equations have the same a.

So, to remove a,

write the equations like this

and subtract the equations.

Elimination Method

a are cancelled.

11d - 7d = 4d

13 - 5 = 8

Divide both sides by 4.

Then d = 2.

d = 2

Put this into a + 7d = 5.

Then a + 7⋅2 = 5.

Substitution Method

+7⋅2 = +14

Move +14 to the right side.

Then a = -9.

a = -9

d = 2

Then a_{n} = -9 + (n - 1)⋅2.

+(n - 1)⋅2 = +2n - 2

-9 + 2n - 2 = 2n - 11

So a_{n} = 2n - 11.