How do you find the average of an arithmetic progression?

To find the average of an arithmetic progression, you first find the sum of the series and then divide this sum by the number of terms.

Understanding Arithmetic Progressions

An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. For example, 2, 4, 6, 8,... is an arithmetic progression where the common difference is 2.

Calculating the Average

Here's how to find the average:

Find the Sum (S) of the Arithmetic Progression:

For an arithmetic progression from 1 to 'n' with a common difference of 1, the formula for the sum 'S' is often given as:

S = n(n + 1) / 2

*   However, if the common difference is not 1, a more general formula for the sum is:

```
S = n/2 * [2a + (n-1)d]
```
Where:

*   'n' is the number of terms.
*   'a' is the first term.
*   'd' is the common difference.

Divide the Sum (S) by the Number of Terms (n):
- Once you have calculated the sum (S), simply divide it by the total number of terms (n) to get the average.
```
    Average = S / n
```

Example:

Let's consider the arithmetic progression from 1 to 100 as in the provided reference.

Calculate the Sum (S):
- Here, n = 100. Using the formula S = n(n + 1) / 2
```
S = 100(100 + 1) / 2
S = 100(101) / 2
S = 50 * 101
S = 5050
```
Calculate the Average:
- Divide the sum by the number of terms.
```
Average = S / n
Average = 5050 / 100
Average = 50.5
```

Therefore, the average of the arithmetic progression from 1 to 100 is 50.5.

Key Insights:

The average of an arithmetic progression is also the median if the number of terms is odd, or it's the average of the two middle terms if the number of terms is even.
The average can be directly calculated as the average of the first and the last term: (first_term + last_term)/2. This works for any arithmetic progression.

Summary Table

Step	Formula	Description
1. Find the Sum (S)	S = n/2 * [2a + (n-1)d] or S = n(n+1)/2	Calculate the sum of all terms in the arithmetic progression. Use S = n(n+1)/2 for AP with a common difference of 1, starting from 1.
2. Calculate the Average	Average = S / n	Divide the calculated sum (S) by the total number of terms (n) to obtain the average.

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