Introduction:

To find the value of "n" such that the standard deviation (SD) of the first "n" natural numbers is 2, we need to understand the formula for calculating the standard deviation and apply it to the given scenario.

Standard Deviation Formula:

The standard deviation (SD) is a statistical measure that quantifies the amount of variation or dispersion in a dataset. The formula for calculating the standard deviation for a set of "n" numbers is as follows:

SD = √(Σ(x - μ)² / n)

Where:
- SD represents the standard deviation
- Σ denotes the summation (sum of)
- x represents each individual number in the dataset
- μ represents the mean (average) of the dataset
- n represents the total number of values in the dataset

Applying the Standard Deviation Formula:

In this case, we are looking for the value of "n" for which the standard deviation is 2. Let's calculate the standard deviation for the first "n" natural numbers and equate it to 2.

SD = 2

Using the formula mentioned above, we can calculate the mean of the first "n" natural numbers:

μ = (1 + 2 + 3 + ... + n) / n = (n(n+1))/2n = (n+1)/2

Now, we can substitute the mean value into the standard deviation formula:

2 = √(Σ(x - (n+1)/2)² / n)

Squaring both sides of the equation, we get:

4 = Σ(x - (n+1)/2)² / n

Breaking down the equation:

To simplify the equation further, let's expand the summation term:

4 = [(1 - (n+1)/2)² + (2 - (n+1)/2)² + (3 - (n+1)/2)² + ... + (n - (n+1)/2)²] / n

Observations:

- The first term in the summation is (1 - (n+1)/2)², which simplifies to [(3 - n)² / 4].
- The second term is (2 - (n+1)/2)², which simplifies to [(2 - n)² / 4].
- Continuing this pattern, the nth term will be [(1 - n)² / 4].
- The sum of these terms can be calculated using the formula for the sum of squares of natural numbers: Σ(n²) = (n(n+1)(2n+1))/6.

Equation Simplification:

Now, we can rewrite the equation as:

4 = [(3 - n)² + (2 - n)² + (1 - n)² + ... + (1 - n)²] / 4n

Simplifying further:

16n = [3² + 2² + 1² + ... + 1²] - 2n[3 + 2 + 1 + ... + 1]

16n = [Σ(n²)] - 2n[Σ(n)]

Using the formula for the sum of squares of natural numbers and the formula for the sum of natural