Question

If mean of first \(n\) odd natural Integer is \(n\) then \(n\) is (a) 2 (b) 3 (c) 1 (d) any natural integer

Short answer

The correct answer is \( (d) \) any natural integer.

Step-by-step solution

Write down the first n odd natural integers

To find the first n odd natural integers, we can represent them as: 1, 3, 5, ... , (2n - 1)

Calculate the sum of the first n odd natural integers

The sum of an arithmetic series is given by: Sum = (number of terms / 2) × (first term + last term) In our case, the number of terms = n, first term = 1, last term = 2n - 1. So, Sum = (n / 2) × (1 + (2n - 1)) Simplifying the equation, Sum = (n / 2) × (2n) Sum = n^2

Calculate the mean of the first n odd natural integers

The mean is given by the sum of the first n odd natural integers divided by the number of integers, which is n. Mean = (Sum of first n odd natural integers) / n Mean = (n^2) / n Mean = n

Compare the obtained mean with the given possible values of n

We have obtained the mean as n. Now we will check if the mean is equal to any of the given options. Option (a): 2 Option (b): 3 Option (c): 1 Option (d): any natural integer We know that the mean is equal to the value of n itself. This means that any natural integer could be a possible value for n.

Choose the correct option

Since the mean of the first n odd natural integers is equal to any natural integer value of n, the correct answer is: Option (d): any natural integer

Key concepts

Odd Natural Numbers

Odd natural numbers are a sequence of numbers that aren't divisible by 2. They are the numbers we encounter in the form of \(1, 3, 5, 7, \) and so on. They are called "natural" because they occur naturally in our counting system. In our problem, these numbers follow a simple pattern: for any integer \(n\), the \(n^{th}\) odd number is given by the formula \(2n - 1\).
This ease of finding the nth odd number is crucial when solving problems involving sequences, such as calculating sums or averages. When tackling a series with odd numbers, first identify the numerical pattern, then apply it for a systematic calculation approach.

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. In the series of odd natural numbers, the common difference is \(2\). This means every number in the series increases by 2 from the previous one, hence maintaining a predictable progress.

• For example, starting at \(1\), the series of odd numbers is \(1, 3, 5,\ldots \)
• The common difference is \(3 - 1 = 2\), \(5 - 3 = 2\), demonstrating the uniform increase.

Understanding this property makes it easier to compute the sum of terms or find specific terms in the series. Using these sequences, one can effortlessly delve into calculations of sums and other related operations.

Sum of Series

The sum of a series refers to the total when all terms are added together. For an arithmetic series, there's a standard formula which simplifies this task: \[\text{Sum} = \frac{\text{number of terms}}{2} \times (\text{first term} + \text{last term})\]Applying this to odd numbers, the formula becomes:

• Number of terms = \(n\)
• First term = \(1\)
• Last term = \(2n - 1\)

Therefore, the sum of the first \(n\) odd numbers is \[\text{Sum} = \frac{n}{2} \times (1 + (2n-1)) = n^2\]This formula reveals a beautiful property of odd numbers; the sum is always a perfect square corresponding to the number of terms.

Mean Calculation

Calculating the mean, or average, of a set of numbers involves finding their sum and dividing by the count of those numbers. In our scenario with the first \(n\) odd natural numbers, we've already determined their sum is \(n^2\).
Thus, the mean is:\[\text{Mean} = \frac{\text{Sum}}{n} = \frac{n^2}{n} = n\]This result shows that the mean of the first \(n\) odd numbers equals \(n\) itself. This is a unique characteristic of the sequence of odd natural numbers. By comprehending this property, one can apply it to verify problems and ensure calculations in arithmetic series are precise and intuitive.