Calculating the Mean of Natural Numbers in an Interval

The question asks us to find the mean of the natural numbers that fall within the interval [15, 64]. This means we need to consider all whole numbers starting from 15 up to and including 64.

Understanding the Interval

The interval notation [15, 64] signifies a closed interval, meaning both the starting number (15) and the ending number (64) are included in the set of numbers we need to consider. The natural numbers in this interval are 15, 16, 17, ..., 63, 64.

What is the Mean?

The mean (or average) of a set of numbers is calculated by summing all the numbers in the set and then dividing the total sum by the count of numbers in the set.

The formula for the mean is:

\(\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total count of numbers}}\)

Finding the Numbers and Their Count

The numbers are a sequence of consecutive natural numbers starting from 15 and ending at 64.

• First number: 15
• Last number: 64

To find the total count of numbers in this sequence, we can use the formula for the number of terms in an arithmetic progression:

\(\text{Count} = \text{Last number} - \text{First number} + 1\)

Let's calculate the count:

\(\text{Count} = 64 - 15 + 1 = 49 + 1 = 50\)

So, there are 50 natural numbers in the interval [15, 64].

Calculating the Mean of Consecutive Numbers

For a sequence of consecutive numbers (an arithmetic progression) like this, the mean can be calculated in two ways:

1. Find the sum of all numbers and divide by the count.
2. Take the average of the first and last number.

Method 1: Sum and Divide

The sum of an arithmetic progression is given by:

\(S = \frac{\text{Count}}{2} (\text{First number} + \text{Last number})\)

Let's calculate the sum:

\(S = \frac{50}{2} (15 + 64) = 25 \times 79\)

Now, calculate the product:

\(25 \times 79 = 1975\)

The sum of natural numbers from 15 to 64 is 1975.

Now, calculate the mean:

\(\text{Mean} = \frac{1975}{50}\)

\(\text{Mean} = \frac{197.5}{5}\)

\(\text{Mean} = 39.5\)

Method 2: Average of First and Last Term

This method is simpler for consecutive numbers. The mean is the average of the first and last term:

\(\text{Mean} = \frac{\text{First number} + \text{Last number}}{2}\)

Let's calculate using this method:

\(\text{Mean} = \frac{15 + 64}{2}\)

\(\text{Mean} = \frac{79}{2}\)

\(\text{Mean} = 39.5\)

Both methods give the same result. The mean of the natural numbers in the interval [15, 64] is 39.5.

Conclusion

The mean of the natural numbers in the interval [15, 64] is 39.5.

Revision Table: Mean of Consecutive Natural Numbers

Additional Information: Arithmetic Progressions and Mean

A sequence of numbers where the difference between consecutive terms is constant is called an arithmetic progression (AP). The natural numbers in the interval [15, 64] (15, 16, 17, ..., 64) form an AP with a common difference of 1.

• In an AP, the mean is always equal to the median if the number of terms is odd.
• If the number of terms is even (like in our case, 50 terms), the mean is exactly halfway between the two middle terms. The two middle terms here would be the 25th and 26th terms. The 25th term is \(15 + (25-1) \times 1 = 15 + 24 = 39\). The 26th term is \(15 + (26-1) \times 1 = 15 + 25 = 40\). The mean is \(\frac{39 + 40}{2} = \frac{79}{2} = 39.5\). This confirms our result.
• Understanding arithmetic progressions simplifies calculations involving sequences of consecutive numbers, including finding their sum and mean.