1. Identify the sequence

The numbers are $8, 21, 34, 47, \ldots, 320$ which form an arithmetic progression (A.P.) with:

• first term: $a=8$
• common difference: $d=13$
• last term: $l=320$

2. Find the number of terms

For an A.P., $l=a+(n-1)d$ So, $320=8+(n-1)13$ $312=13(n-1)$ $n-1=24$ $n=25$

Thus, there are $25$ terms.

3. Use variance formula for an A.P.

The terms can be written as $a, a+d, a+2d, \ldots, a+(n-1)d$ Adding a constant $a$ does not change variance, so the variance of the A.P. is: $d^2 \times \text{Var}(0,1,2,\ldots,n-1)$

Now, $\text{Var}(0,1,2,\ldots,n-1)=\frac{n^2-1}{12}$

Hence, $\sigma^2=d^2\cdot \frac{n^2-1}{12}$

Substitute $d=13$ and $n=25$: $\sigma^2=13^2\cdot \frac{25^2-1}{12}$ $=169\cdot \frac{625-1}{12}$ $=169\cdot \frac{624}{12}$ $=169\cdot 52$ $=8788$

4. Final answer

The variance is $\boxed{8788}$

5. Comparison with stored answer

Stored correct answer: $8788$

This matches the derived answer.