An arithmetic sequence is formed by repeatedly adding a constant value, called the common difference, to the previous term.
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between consecutive terms remains constant. This constant difference is known as the "common difference," often denoted as 'd'.
General Form
The general form of an arithmetic sequence is:
a, a + d, a + 2d, a + 3d, a + 4d, ... , a + (n-1)d
Where:
- a is the first term of the sequence.
- d is the common difference between consecutive terms.
- n is the number of terms in the sequence.
Key Elements
- First Term (a): This is the starting point of the sequence.
- Common Difference (d): This is the constant value added to each term to get the next term. It can be positive, negative, or zero.
Example
Consider the arithmetic sequence: 2, 5, 8, 11, 14, ...
- The first term (a) is 2.
- The common difference (d) is 3 (because 5-2 = 3, 8-5 = 3, and so on).
How to Construct an Arithmetic Sequence
- Choose a first term (a). This can be any number.
- Choose a common difference (d). This can also be any number.
- Generate subsequent terms: Add the common difference (d) to the previous term to get the next term. Repeat this process to create the desired number of terms.
For example, if a = 7 and d = -2, the sequence would be: 7, 5, 3, 1, -1, ...
Identifying an Arithmetic Sequence
To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. If the difference is the same throughout the sequence, then it is an arithmetic sequence.
