Arithmetic Sequences

Arithmetic Sequences

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

Formula

An arithmetic sequence is a sequence whose differences of the adjacent terms are the same. an = a1 + (n - 1)d. an: nth term, a1: First term, d: Common difference

An arithmetic sequence is a sequence
whose differences of the adjacent terms
are the same (= d).

a2 - a1 = d
a3 - a2 = d
a4 - a3 = d
...

d is the [common difference].

So to find the next term,
add the common difference: +d.

The nth term of an arithmetic sequence, an,
can be found by using the formula below.

an = a1 + (n - 1)d

an: nth term
a1: First term
d: Common difference

Example 1

For the given arithmetic sequence, find an. 1, 4, 7, 10, 13, ...

The first term is 1.
So a1 = 1.

4 - 1 = 3
7 - 4 = 3
10 - 7 = 3
13 - 10 = 3
...
So d = 3.

So an = 1 + (n - 1)⋅3.

+(n - 1)⋅3 = +3n - 3

1 - 3 = -2

So an = 3n - 2.

Example 2

An arithmetic sequence is given below. -2, 5, 12, 19, ... If ak = 551, find the value of k.

The first term is -2.
So a1 = -2.

5 - (-2) = 7
12 - 5 = 7
19 - 12 = 7
...
So d = 7.

So ak = -2 + (k - 1)⋅7.

+(k - 1)⋅7 = +7k - 7

-2 - 7 = -9

So ak = 7k - 9.

ak = 7k - 9

And it says
ak = 551.

So 7k - 9 = 551.

Move -9 to the right side.

Then 7k = 560.

Divide both sides by 7.

Then k = 80.

Example 3

For an arithmetic sequence, a8 = 5 and a12 = 13. Find an.

a8 = a1 + 7d = 5
a12 = a1 + 11d = 13

So
a1 + 7d = 5,
a1 + 11d = 13.

The goal is to find a1 and d.

a1 + 11d = 13
a1 + 7d = 5

Subtract these two equations
to eliminate a1.

Then 4d = 8.

Elimination method

Divide both sides by 4.

Then d = 2.

Put [d = 2]
into [a1 + 7d = 5].

Then a1 + 7⋅2 = 5.

+7⋅2 = +14

Move +14 to the right side.

Then a1 = -9.

a1 = -9
d = 2

So an = -9 + (n - 1)⋅2.

+(n - 1)⋅2 = +2n - 2

-9 - 2 = -11

So an = 2n - 11.