The average (arithmetic mean) of the positive integers from 1 to 100 inclusive is 50.5.
Here's how to calculate it:
Understanding Arithmetic Mean
The arithmetic mean is the sum of a set of numbers divided by the count of those numbers. In simpler terms, it's the "average."
Calculation
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Sum of integers from 1 to 100: The sum of the first n positive integers can be calculated using the formula: n(n+1)/2. In this case, n = 100.
- Sum = 100 * (100 + 1) / 2
- Sum = 100 * 101 / 2
- Sum = 10100 / 2
- Sum = 5050
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Number of integers: There are 100 positive integers from 1 to 100 inclusive.
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Arithmetic Mean: Divide the sum by the number of integers.
- Average = 5050 / 100
- Average = 50.5
Alternative Explanation (Using Properties of Arithmetic Progressions)
The sequence of integers from 1 to 100 forms an arithmetic progression with a common difference of 1. The average of an arithmetic progression is simply the average of the first and last terms.
- Average = (First term + Last term) / 2
- Average = (1 + 100) / 2
- Average = 101 / 2
- Average = 50.5
Therefore, the average arithmetic mean of the positive integers from 1 to 100 inclusive is 50.5.
