Arithemetic Sequences

Sequences.jpgAn arithmetic sequence or progression, is a sequence where each term is calculated by adding a fixed amount to the previous term.

Notation

This fixed amount is called the common difference, d. It can be positive or negative.

The common difference can be calculated by subtracting a term from the one following it. 
e.g. Common difference = t n+1 − t n

The first term of an arithmetic sequence is shown by the variable a.

Arithmetic Sequence
First term, a
Common difference, d
2, 6, 10, 14, ...
2

6 − 2 = 4
10 − 6 = 4        etc.
20, 14, 8, 2, ...
20
14 − 20 = -6
8 − 14 = -6       etc.

General Term, tn

An arithmetic sequence can be written:

First term
Second term
Third term
Fourth term
  
General term (n th term)
t 1
t 2
t 3
t 4
...
tn
a
a + d
a + 2d
a + 3d
...
a + (n − 1)d

 

Example 1

What is the common difference of the arithmetic sequence:

3.5, 8.3, 13.1, 17.9, ...

Common difference = t n+1 − t n

Second term − first term = t2 − t1 = 8.3 − 3.5 = 4.8

Check:

Third term − second term = 13.1 − 8.3 = 4.8

The common difference is 4.8
Example 2

Find the 20th term of the arithmetic sequence:

3, 9, 15, 21, ...

Common difference, d = 9 − 3 = 6
First term a = 3

using tn = a + (n − 1)d

t20 = 3 + (20 − 1)6
= 3 + 19 x 6
= 117

The 20th term is 117
Example 3 Which term of the sequence 12, 15, 18, .. would be equal to 54?

Common difference, d = 3
First term, a = 12

Using tn = a + (n − 1)d

54 = 12 + (n − 1)3
54 = 12 + 3n − 3
54 = 3n + 9
3n = 45
n = 15

The 15th term would be 54
Example 4

The third term of an arithmetic sequence is 18 and the tenth term is 74.

Find the first term, a, and the common difference, d, and thus list the first four terms of the sequence.

t 3 = 18
t10 = 74

using tn = a + (n − 1)d

18 = a + (3 − 1)d
74 = a + (10 -1)d

18 = a + 2d
74 = a + 9d

56 = 7d              ( subtracting)
d = 8                 the common difference

18 = a + 2 x 8
a = 2                 the first term

The sequence is 2, 10, 18, 26, ...