sequences and series - A problem about the ratio of atoms

I'm trying to find the ratio of the number of atoms in the core of a nanoparticle, and the outermost shell with each shell containing:

$10k^2+2$ atoms

Then the total number up to the $k^{th}$ shell is (I believe)

$\sum\limits_{n=1}^{k-1}(10(k-n)^2+2)$

It's this expression I'm stuck on. I'm not 100% on what I'm allowed to do to evaluate this summation. I couldn't find much about operations on summations when I went looking.

Is this expression equivalent?

$10\bigg(\sum\limits_{n=1}^{k-1}k^2+\sum\limits_{n=1}^{k-1}n^2-\sum\limits_{n=1}^{k-1}2kn\bigg)+\sum\limits_{n=1}^{k-1}2$

If so how can I evaluate these?

Edit: Note, errors fixed thanks to Gerry's advice in his answer

Answer

Remember $(k-n)^2=k^2+n^2-2kn$, so you're missing a factor of 2. Also, the 10 doesn't multiply the +2 in the sum, so your last sum shouldn't be in the parentheses.

But you are making it a bit harder than it has to be. Do you see that

$$\sum_{n=1}^{k-1}(10(k-n)^2+2)=\sum_{n=1}^{k-1}(10n^2+2)$$ On the left, as $n$ goes from $1$ to $k-1$, $k-n$ goes from $k-1$ to $1$, so the form on the right adds up the same numbers.

Now all you need is the formula for $\sum_{n=1}^{k-1}n^2$, which you can find in any number of places. .