To find the arithmetic mean of an arithmetic progression, you can use a few different methods, depending on the information you have available. The most straightforward method involves understanding what the arithmetic mean is. As the provided reference states, the arithmetic mean, or mean of the group, is the sum of all the numbers in a group divided by the number of items in that list.
Here's a breakdown of the methods:
1. Using the Sum of the Progression
If you know the sum of the arithmetic progression and the number of terms, the arithmetic mean is simply the sum divided by the number of terms.
Formula:
Arithmetic Mean = (Sum of the terms) / (Number of terms)
Example:
Suppose an arithmetic progression has 5 terms and their sum is 25. The arithmetic mean is 25 / 5 = 5.
2. Using the First and Last Term
In an arithmetic progression, the arithmetic mean is also equal to the average of the first and last terms. This is a quick and easy method when you know these two values.
Formula:
Arithmetic Mean = (First Term + Last Term) / 2
Example:
Consider the arithmetic progression: 2, 4, 6, 8, 10. The first term is 2 and the last term is 10. Therefore, the arithmetic mean is (2 + 10) / 2 = 6.
3. Using Any Two Terms Equidistant from the Ends
More generally, the arithmetic mean is the average of any two terms that are equidistant from the beginning and the end of the progression.
Example:
In the sequence 1, 3, 5, 7, 9, choose the 2nd and 4th term (3 and 7 respectively). These are equally distant from the beginning and end. Their average is (3+7)/2 = 5, which is the arithmetic mean.
Summary Table
| Method | Formula | Requirements |
|---|---|---|
| Sum and Number of Terms | (Sum of terms) / (Number of terms) | Sum of terms, number of terms |
| First and Last Term | (First Term + Last Term) / 2 | First term, last term |
| Equidistant Terms | (Term i + Term n-i+1) / 2 | Two terms equidistant from start and end |
