2. If you already know Faulhaber's formula, which says
then it's just a matter of setting
. If you don't, then you can prove that it works (via induction), or compute the sum by some other means. Presumably you're not expected to use brute force and just add the squares of 1 through 4.
Just to demonstrate one possible method of verifying the formula, suppose we start from the binomial expansion of
, do some manipulation, then sum over 
:
The left side is a telescoping series - several terms in consecutive terms of the series will cancel - and reduces to
. For example,
Distributing the sum on the right side across each term and pull out constant factors to get
If you don't know the formula for
, you can use a similar trick with the binomial expansion 
, or a simpler trick due to Gauss, or other methods. I'll assume you know it to save space for the other parts of your question. We then have
and when we get 30.
3. Each term in the sum is a cube, but the sign changes. Recall that
is either 1 if 
is even or -1 if is odd. So we can write
(
as the exponent to -1 also works)
6. If
, and 
, then we would get 
. So the sum with respect to 
is
then it's just a matter of setting
Just to demonstrate one possible method of verifying the formula, suppose we start from the binomial expansion of
The left side is a telescoping series - several terms in consecutive terms of the series will cancel - and reduces to
Distributing the sum on the right side across each term and pull out constant factors to get
If you don't know the formula for
and when
3. Each term in the sum is a cube, but the sign changes. Recall that
(
6. If