A sequence of numbers where successive terms differ by a constant is called an arithmetic sequence. According to the provided reference, an arithmetic sequence is a sequence in which the difference between consecutive terms remains the same.
Understanding Arithmetic Sequences
Arithmetic sequences are characterized by a constant difference between adjacent terms. This constant difference is often referred to as the "common difference".
- Common Difference: The value obtained by subtracting any term from its succeeding term in an arithmetic sequence.
Examples of Arithmetic Sequences
Here are a few examples to illustrate the concept:
- Example 1: 2, 4, 6, 8, 10... (Common difference = 2)
- Example 2: 1, 5, 9, 13, 17... (Common difference = 4)
- Example 3: 10, 7, 4, 1, -2... (Common difference = -3)
Formula for the nth term of an Arithmetic Sequence
The nth term of an arithmetic sequence can be calculated using the following formula:
an = a1 + (n - 1)d
Where:
- an is the nth term.
- a1 is the first term.
- n is the term number.
- d is the common difference.
Practical Insights
- Arithmetic sequences are fundamental in various areas of mathematics and have real-world applications, such as simple interest calculations, evenly spaced seating arrangements, and patterns observed in nature.
- Identifying the common difference is key to understanding and working with arithmetic sequences.
