Successive terms differ by a constant in what is known as an arithmetic sequence.
Understanding Arithmetic Sequences
An arithmetic sequence is a special type of sequence where the difference between any two consecutive terms is always the same. This consistent difference is called the common difference.
Definition
According to our reference, an arithmetic sequence has the form:
{an} = {a1, a1+d, a1+2d, a1+3d,...}
Where:
- a1 is the first term of the sequence.
- d is the common difference.
- an represents the nth term of the sequence.
Key Characteristics
- Constant Difference: The hallmark of an arithmetic sequence is the constant difference between each term.
- Linear Progression: The terms progress in a linear manner, either increasing or decreasing consistently.
- Simple Formula: A simple formula allows us to calculate any term in the sequence given the first term and the common difference.
Examples
Let's look at some examples of arithmetic sequences:
- Example 1: Increasing Sequence
- Sequence: 2, 5, 8, 11, 14...
- a1 = 2
- d = 3 (each term is 3 more than the previous)
- Example 2: Decreasing Sequence
- Sequence: 10, 7, 4, 1, -2...
- a1 = 10
- d = -3 (each term is 3 less than the previous)
Practical Applications
Arithmetic sequences are found in many real-world situations, such as:
- Simple Interest: The accumulated interest over regular intervals with a fixed interest rate, grows in an arithmetic manner.
- Depreciation: The value of an asset decreases in a linear manner over time.
- Stacking: When objects are stacked in a uniform way, the number of objects in each row can form an arithmetic sequence.
Key Takeaway
In summary, successive terms differ by a constant in a mathematical sequence known as an arithmetic sequence. This constant difference, or common difference, is the defining characteristic of such sequences. The form of an arithmetic sequence is based on its first term and common difference and can be represented by a simple formula.
