The arithmetic mean of an arithmetic progression is simply the average of the terms in the progression.
Here's a more detailed explanation:
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.
Understanding Arithmetic Mean in an AP
If you have an arithmetic progression, the arithmetic mean can be calculated in a few ways, all of which are equivalent.
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General Definition: The arithmetic mean is the sum of the terms divided by the number of terms. If S is the sum of N terms in the AP, then the arithmetic mean, A, is:
A = S / N
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For a Simple Sequence of Three Terms: If a, b, and c form an AP, then b is the arithmetic mean of a and c, and it can be calculated as:
b = (a + c) / 2
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Using the First and Last Term: In any finite arithmetic progression, the arithmetic mean is equal to the average of the first term (a1) and the last term (an):
Arithmetic Mean = (a1 + an) / 2
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Using the Middle Term: If the arithmetic progression contains an odd number of terms, the arithmetic mean is simply the middle term.
Example
Consider the arithmetic progression: 2, 4, 6, 8, 10
- Sum of Terms: 2 + 4 + 6 + 8 + 10 = 30
- Number of Terms: 5
- Arithmetic Mean: 30 / 5 = 6
Alternatively, using the first and last term:
- First Term: 2
- Last Term: 10
- Arithmetic Mean: (2 + 10) / 2 = 6
In this case, the middle term is also 6.
In summary, the arithmetic mean of an arithmetic progression represents the average value of its terms and can be easily calculated using various methods, most notably by averaging the first and last terms or summing all terms and dividing by the total number of terms.
