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.

a1 + d = a2
a2 + d = a3
a3 + d = a4
...

This sequence is an arithmetic sequence.

This d is called the common difference.
It is the difference of the adjacent terms.
(a2 - a1 = d, a3 - a2 = d, ...)

Formula

Formula

For an arithmetic sequence,
an = a + (n - 1)d.

an: nth term
a: First term, a1
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 an = 1 + (n - 1)⋅3.

+(n - 1)⋅3 = +3n - 3

Multiply a Monomial and a Polynomial

1 + 3n - 3 = 3n - 2

So an = 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 ak = -2 + (k - 1)⋅7.

+(k - 1)⋅7 = +7k - 7

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

So ak = 7k - 9.

It says
ak = 551.

And you found
ak = 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 an,
find the first term a
and the common difference d.

For an arithmetic sequence,
a8 = a + 7⋅d.

And it says
a8 = 5.

So a8 = a + 7d = 5.

For an arithmetic sequence,
a12 = a + 11⋅d.

And it says
a12 = 13.

So a12 = 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 an = -9 + (n - 1)⋅2.

+(n - 1)⋅2 = +2n - 2

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

So an = 2n - 11.