1. Given data

We need the variance of the numbers $1,2,3,\dots,n$ to be $14$, where $n$ is an odd natural number.

2. Mean of the first $n$ natural numbers

The mean is $\bar{x}=\frac{1+2+3+\cdots+n}{n}=\frac{\frac{n(n+1)}{2}}{n}=\frac{n+1}{2}.$

3. Variance formula

Variance is $\sigma^2=\frac{1}{n}\sum_{k=1}^{n}(k-\bar{x})^2.$

For the numbers $1,2,\dots,n$, a standard result is $\text{Var}(1,2,\dots,n)=\frac{n^2-1}{12}.$

So we set $\frac{n^2-1}{12}=14.$

4. Solve for $n$

$n^2-1=168$ $n^2=169$ $n=13$

Since $n$ is a natural number (and odd), $n=13.$

5. Verification with stored answer

Our derived answer is $13$. The stored correct answer is also $13$.

So they agree.