Arithmetic Sequences

Arithmetic Sequences

How to solve the arithmetic sequence problems: formula, proof, 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).

an = a1 + (n - 1)d

an: nth term
a1: first term
d: common difference

Proof

Arithmetic Sequences: Proof of the Formula

If the common difference is d, then

a1 = a1
a2 = a1 + 1d
a3 = a1 + 2d
...
an = a1 + (n - 1)d.

Example 1

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

Find a1 and d.

a1 = 1, d = +3

a1 = 1, d = +3

an = 1 + (n - 1)⋅3
= 3n - 2

Example 2

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

Find a1 and d.

a1 = -2, d = +7

a1 = -2, d = +7

an = -2 + (n - 1)⋅7
= 7n - 9

ak = 7k - 9

And it says ak = 551.

So ak = 7k - 9 = 551.

k = 80

Example 3

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

Write a8 and a12
by using a1 and d.

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

The goal is to find a1 and d.

Use the elimination method to solve this system.

Then d = 2.

Elimination method

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

Then a1 = -9.

a1 = -9, d = 2

So an = -9 + (n - 1)⋅2
= 2n - 11