Arithmetic sequences
In a nutshell
An arithmetic sequence is one where there is a common difference between each term.
Using arithmetic sequences
The common difference of an arithmetic sequence can be used to find any term in the sequence. The nth term, un, of an arithmetic sequence can be given by an equation where a is the first term and d is the common difference.
un=a+(n−1)d
Note: The common difference can be positive or negative.
Example 1
An arithmetic sequence is given by the formula
un=5−8(n−1)
What is the first term of the sequence and what is the common difference between each term? Hence give the first four terms in the sequence.
By comparing to the general formula above, you have that the first term is 5 and that the common difference is −8.
Thus the first four terms are:
5,−3,−11,−19
Example 2
The fifth term of an arithmetic sequence is 8 and the ninth term is 23. What is the first term, a, and the common difference, d? Find the 101th term.
Express the fifth and ninth term algebraically:
5th term:u59th term:u9=a+(5−1)d=a+4d=8=a+(9−1)d=a+8d=23
Solve the equations simultaneously:
a+8da+4d4d=23=8=15
d=3.75
Solve for a:
a+4(3.75)a+15=8=8
a=7
Find the 101th term:
u101=a+(n−1)d=7+(n−1)(3.75)
=7+(101−1)(3.75)=7+375
101th term: u101=382