I was trying to help some friends with their math homework, and I am completely stumped. Now I am curious myself on how to solve it:
Determine the following sum, in terms of n: Summation from i = 0 to n, of (-1)^k * (n choose k) * 10^k
These are all of the details given for the problem. I have to assume that k is just some arbitrary constant, but even that is not given. Is it possible to eliminate k entirely from the result? Or did the teacher just make a mistake, where they should have said "Determine the following sum, in terms of n and k"?
Presumably the teacher intended $k$ to be the summation index.
Note that (assuming $i$ and $k$ are different) there is no $i$ in your sum, so as it stands the sum is just $(n+1)\times (-1)^k\times \binom nk\times 10^k$.
Of course, if $k=i$, the summation index, then this sum becomes a simple case of the binomial theorem.