Math 114 Arithmetic Sequences and Series
The follow examples of numbers are called arithmetic sequences. You add the same number
each time to get the next number in the sequence. Each number in the sequence is called a term.
Some examples of arithmetic sequences:
Example 1: 1, 2, 3, 4, 5, 6, …, 100
Example 3: -4, 1, 6, 11, 16, 21, …
Example 2: 3, 7, 11, 15, 19, 23, …
An arithmetic sequence of numbers has a common difference. It is the number you add each
time to get the next number in the sequence. We call the common difference d .
In Example 1, d  1. In Example 2: d  4 . In Example 3, d  5 .
We can represent the first term in the sequence by a1 , the 2nd term by a2 and the 3rd term by a3 .
The nth term is an . In Example 3 above, a1  4 a2  1 a3  6 a4  11 and so on.
We can write any sequence of numbers as a1 , a2 , a3 ,..., an . This is called a finite sequence
because it has a particular number of terms. Example: -10, -5, 0, 5, 10, 15, 20 There are 7
terms so n = 7. A sequence that is not finite is infinite. We cannot count the terms in an infinite
sequence. Example: -10, -5, 0, 5, 10, 15, 20, 25, 30, 35, … (Dots mean the terms go on and
on.)
To get any term in an arithmetic sequence, use the formula
an  a1  (n  1)d
Example: Find the 40th term of the sequence with first 3 terms 8, 11, 14.
n  40, a1  8, d  3 .
Solution:
a40  a1  (40  1)d
a40  8  (40  1)3  125
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Arithmetic Series: You add the terms in an arithmetic sequence to get an arithmetic series.
Example (finite sequence): 10  5  0  5  10  15  20 .
Example (infinite sequence): 10  5  0  5  10  15  20  ...
To get the sum of the first n terms of an arithmetic series, use the formula
Sn 
n
 a1  an 
2
Example: Find the sum of the first 40 terms of the sequence with first 3 terms 8, 11, 14.
n  40, a1  8 .
an  a40  125 from what we calculated above.
Solution:
40
40
S40   a1  a40   8  125  2660
2
2