The Mean of the First n Odd Natural Numbers

To solve this problem, we need to find the value of n for which the mean of the first n odd natural numbers is equal to n itself.

Understanding the Mean

The mean of a set of numbers is calculated by adding up all the numbers in the set and then dividing the sum by the total number of values in the set.

Calculating the Mean of Odd Natural Numbers

The first n odd natural numbers can be written as 1, 3, 5, 7, ..., (2n-1). To find their mean, we add up all these numbers and divide by n.

Sum of the first n odd natural numbers = 1 + 3 + 5 + 7 + ... + (2n-1)

To calculate the sum of an arithmetic series, we use the formula: Sn = (n/2)(a + l), where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.

In this case, the first term is 1 and the last term is (2n-1). So, the sum of the first n odd natural numbers can be represented as:

Sn = (n/2)(1 + (2n-1))

Simplifying the expression:

Sn = (n/2)(1 + 2n - 1)
= (n/2)(2n)
= n^2

The mean of the first n odd natural numbers is given by:

Mean = Sn/n = n^2/n = n

Determining the Value of n

According to the given condition, the mean of the first n odd natural numbers is equal to n itself. Therefore, we have:

n = n

This equation holds true for all natural numbers. Hence, the correct answer is option B - any natural number.

Conclusion

The mean of the first n odd natural numbers is equal to n itself for any natural number. This can be mathematically proven by calculating the sum of the series and dividing it by n. Therefore, the correct answer to the given question is option B.