The commonality between successive terms in an arithmetic sequence is a constant difference.
Understanding Arithmetic Sequences
An arithmetic sequence is a series of numbers where the difference between any two consecutive terms remains the same. This constant difference is called the "common difference". As referenced, "an arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term".
Key Features:
- Constant Difference: The core characteristic is the consistent value added (or subtracted) to get from one term to the next.
- Linear Progression: Arithmetic sequences demonstrate a linear pattern of change.
- Predictability: You can predict any term in the sequence if you know the first term and the common difference.
Example of an Arithmetic Sequence:
Let's examine the given example, where the arithmetic sequence is: 3, 9, 15, 21, 27.
Calculating the Common Difference:
We can find the common difference by subtracting any term from its immediate successor:
- 9 - 3 = 6
- 15 - 9 = 6
- 21 - 15 = 6
- 27 - 21 = 6
In this sequence, the common difference is consistently 6.
How it works:
- First Term: The sequence starts at 3.
- Second Term: 3 + 6 = 9
- Third Term: 9 + 6 = 15
- Fourth Term: 15 + 6 = 21
- Fifth Term: 21 + 6 = 27
Significance of the Common Difference
The common difference is the defining characteristic of an arithmetic sequence. It allows:
- Identification: You can quickly identify if a sequence is arithmetic by checking for a constant difference.
- Progression: It provides the rule for generating the next number within the sequence.
- Formula Derivation: It is essential to write the formula of the arithmetic sequence and find the sum of the sequence.
In summary, the common element that links terms in arithmetic sequence is the consistent amount added or subtracted each time; this value is called the common difference, as stated in the reference material.
