A set of numbers with a constant difference between consecutive terms is called an arithmetic progression or arithmetic sequence.
Understanding Arithmetic Progressions
An arithmetic progression is a sequence where the difference between any term and its preceding term remains the same throughout the entire sequence. This constant difference is known as the common difference.
Key Characteristics:
- Constant Difference: The defining feature of an arithmetic progression.
- Terms: Each number in the sequence is a term.
- Common Difference: The value added (or subtracted) to get from one term to the next.
Examples of Arithmetic Progressions:
Here are a few examples to illustrate the concept:
- 2, 4, 6, 8, 10 (Common difference: 2)
- 1, 5, 9, 13, 17 (Common difference: 4)
- 10, 7, 4, 1, -2 (Common difference: -3)
Formula for the nth Term:
The nth term (an) of an arithmetic progression can be calculated using the formula:
an = a1 + (n - 1)d
Where:
- a1 is the first term.
- n is the term number.
- d is the common difference.
Example:
Let's find the 10th term of the arithmetic progression 2, 5, 8, 11,...
- a1 = 2
- d = 3
- n = 10
a10 = 2 + (10 - 1) 3 = 2 + 9 3 = 2 + 27 = 29
Therefore, the 10th term is 29.
Real-World Applications
Arithmetic progressions have many applications in various fields, including:
- Finance: Calculating simple interest.
- Physics: Analyzing uniformly accelerated motion.
- Computer Science: Designing algorithms.
- Everyday Life: Predicting patterns and making estimations.
