Arithmetic Sequence - Formula, Definition, Examples
Arithmetic Sequence - Formula, Definition, Examples
TL;DR
An arithmetic sequence (or arithmetic progression, AP) is an ordered list of numbers where the difference between consecutive terms is constant — that constant is called the common difference (ddd). The nth-term formula is an=a1+(n−1)d and the sum of the first n terms is Sn=n2(a1+an).
What Is an Arithmetic Sequence?
An arithmetic sequence is an ordered list of numbers where each term is obtained by adding a fixed constant to the previous one. That fixed constant is the common difference, denoted d.
Example sequence: 3,7,11,15,19,… — common difference d=4.
The general form is:
a1,;a1+d,;a1+2d,;a1+3d,;…
where a1 is the first term and d is the common difference. A sequence is arithmetic if and only if the difference between every pair of consecutive terms is the same.
What Is the Arithmetic Sequence Formula?
There are two formulas you need.
The nth-Term Formula
The n-th term is given by:
a_n = a_1 + (n - 1) d
where a1 is the first term, d is the common difference, and n is the position of the term you want.
Example. Find the 20th term of 5,8,11,14,… Here a1=5, d=3, n=20:
a_{20} = 5 + (20 - 1)(3) = 62
The Sum Formula (Arithmetic Series)
The sum of the first n terms — called an arithmetic series and written Sn — has two equivalent forms:
S_n = \frac{n}{2}(2a_1 + (n - 1)d)
or equivalently S_n = \frac{n}{2}(a_1 + a_n)
Example. Sum the first 20 terms of 5,8,11,14,… We found a_{20} = 62:
S_{20} = \frac{20}{2}(5 + 62) = 670
How Do You Find the Common Difference?
Subtract any term from the next one:
d = a_2 - a_1 = a_3 - a_2 = …
If the difference is the same every time, the sequence is arithmetic.
What Is the Difference Between Arithmetic and Geometric Progressions?
An arithmetic progression (AP) adds a constant; a geometric progression (GP) multiplies by a constant.
| Feature | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
| Rule | Add a constant each step | Multiply by a constant each step |
| Constant called | Common difference, d | Common ratio, r |
| nth term | a_n = a_1 + (n-1)d | a_n = a_1 \cdot r^{n-1} |
| Sum of first n | S_n = \frac{n}{2}(a_1 + a_n) | S_n = a_1 \cdot \frac{r^n - 1}{r - 1}, r≠1 |
| Growth shape | Linear | Exponential |
Frequently Asked Questions
What is an arithmetic sequence in simple words?
An arithmetic sequence is a list of numbers where each term is obtained by adding the same number to the previous one.
What is the formula for an arithmetic sequence?
The n-th term: a_n = a_1 + (n−1)d, where a_1 is the first term, d is the common difference, and n is the position.
How do you find the common difference?
Subtract any term from the next: d = a_2 - a_1.
What is the difference between arithmetic and geometric sequence?
An arithmetic sequence adds the same number each step (common difference). A geometric sequence multiplies by the same number each step (common ratio).