Verify the Product of a Summation

Question

Can anybody verify that the below equation equals $0$?

$\prod\limits_{k=2}^{10} (\sum\limits_{i=1}^{k-1}(2(i-1)))$

Here is my work, I believe it's correct:

Note: The sequence continues, I just didn't write them all out because anything times $0$ is $0$.

Answer 1

It is correct. I suspect whomever gave problem gave it as a joke in the sense that it looks intimidating but is really simple

Answer 2

You can make it slightly faster, considering the inner summation $$S_k=\sum_{i=1}^{k-1}2(i-1)=2\sum_{i=1}^{k-1} i-2\sum_{i=1}^{k-1} 1=2\times\frac{(k-1) k}{2}-2 (k-1)$$ $$S_k=k^2-3 k+2=(k-1)(k-2)$$ Now, you want to consider $$R=\prod_{k=a}^b S_k=\prod_{k=a}^b (k-1)(k-2)$$ It is clear that if $a=1$ or $a=2$, the result will be zero as you proved it ! In fact, if will be zero for any $a<1$ and in many other cases (as soon as during the product, $k$ would be equal to $1$ or $2$).