Bounding the sum of the first n cubes with rectangles, then finding it exactly by differences and induction

This question explores how the sum of the first \(n\) cubes can first be estimated and then found exactly.

For \(n\in\mathbb{Z}^+\), let \(C(n)=1^3+2^3+3^3+\cdots+n^3\).

1. 1. Calculate \(C(4)\).[1]
   2. Write down an expression, in terms of \(n\), for \(C(n+1)-C(n)\).[1]
2. The positive odd numbers \(1,3,5,\ldots\) form an arithmetic sequence.
   Treating the first \(m\) positive odd numbers as an arithmetic series, show that their total is \(m^2\).[3]
3. Each cube can be split into a sum of consecutive odd numbers: \(1^3=1\), \(2^3=3+5\) and \(3^3=7+9+11\).
   Write \(4^3\) as a sum of consecutive odd numbers that continues this pattern.[1]
4. The graph below shows three rectangles, each of width \(1\), with heights \(1^3\), \(2^3\) and \(3^3\). The pattern can be continued to give as many rectangles as required.
   Each rectangle has its upper right-hand vertex on the curve \(y=g(x)\).
   1. Write down \(g(x)\).[1]
   2. By comparing the combined area of \(n\) such rectangles with a suitable integral, show that \(C(n)>\frac{n^4}{4}\).[4]
   3. By moving every rectangle one unit to the right, or otherwise, show that \(C(n)<\frac{(n+1)^4-1}{4}\).[4]
5. The sequence \(w_n\) is defined by \(w_n=n^2(n+1)^2-(n-1)^2n^2\) for \(n\in\mathbb{Z}^+\). The table gives some values of \(w_n\) and of \(\sum\limits_{r=1}^{n}w_r\).

   \(n\)	\(w_n\)	\(\sum\limits_{r=1}^{n}w_r\)
   1	4	4
   2	32	36
   3	\(a\)	\(b\)
   4	256	400

   1. Find \(a\).[1]
   2. Find \(b\).[1]
6. 1. Suggest an expression, in terms of \(n\), for \(\sum\limits_{r=1}^{n}w_r\).[1]
   2. Prove by mathematical induction that your expression is correct for every \(n\in\mathbb{Z}^+\).[3]
7. 1. Find \(w_n\) in its simplest form.[2]
   2. Hence, write \(\sum\limits_{r=1}^{n}w_r\) in terms of \(C(n)\).[2]
   3. Hence, show that \(C(n)=\frac{n^2(n+1)^2}{4}\).[3]
   4. Verify that the expression for \(C(n)\) in part (g)(iii) satisfies both inequalities in part (d).[2]