Animated Solution for Mathematics - Statistics: If the mean deviation about the mean of the numbers $1,2,3,\dots ,n$, where $n$ is odd, is $\frac{5(n+1)}{n}$, then $n$ is equal to ______.

Imagine you are standing on a long, straight path. Along this path, there are markers placed at positions $1,2,3,\dots ,n$. You are told that $n$ is an odd number.

Your task is to understand the 'spread' of these markers—specifically, their Mean Deviation about the mean. This might sound like a dry statistical exercise, but there is a hidden, elegant geometry waiting to be uncovered.

Finding the Center of Gravity

Before we can measure how far these markers are from the center, we must find the center itself. The mean, denoted as $\bar{x}$, is the arithmetic average of our data set.

The sum of the first $n$ natural numbers is given by the classic formula:

To find the mean, we divide this sum by the total number of terms, $n$. The $n$ in the numerator and the $n$ in the denominator cancel out beautifully, leaving us with:

This point is the 'center of gravity' of our data set. Since $n$ is odd, $\bar{x}$ is an integer, sitting perfectly in the middle of our sequence.

The Dance of Deviations

Now, let us look at the deviations—the distance of each marker from our mean. We are calculating:

Because our data is a simple sequence of integers, the deviations form a symmetric pattern. Let $n=2k+1$. Our mean becomes $k+1$.

The deviations are $|1-(k+1)|,|2-(k+1)|,\dots ,|(2k+1)-(k+1)|$. This results in the sequence $k,k-1,\dots ,1,0,1,\dots ,k-1,k$.

Notice the symmetry? We have two identical sets of numbers: $1,2,\dots ,k$ on both sides of the mean. This symmetry is the key to unlocking the problem without tedious summation.

The Algebraic Bridge

Since we have two identical sets of deviations, the total sum of deviations is simply $2\times {\sum }_{i=1}^{k}i$. The sum of the first $k$ natural numbers is:

Multiplying this by $2$, we get $k(k+1)$. Now, we substitute $k=\frac{n-1}{2}$ back into our expression.

The sum of deviations becomes:

Finally, dividing by $n$ to get the Mean Deviation, we arrive at:

The problem provides us with the value of the Mean Deviation: $\frac{5(n+1)}{n}$. We now equate our derived expression to this given value:

Recognizing that $n^2-1$ is a difference of squares, we factor it into $(n-1)(n+1)$. The equation becomes:

Since $n$ is a natural number, we can safely cancel $n$ and $(n+1)$ from both sides. We are left with $\frac{n-1}{4}=5$, which simplifies to $n-1=20$, or $n=21$.

It is an odd natural number, satisfying all our conditions. The beauty of this problem lies not just in the calculation, but in how symmetry transforms a potentially complex summation into a simple, elegant algebraic identity.