5.
\begin{figure}[h]
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\caption{Figure 2}
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A block has length \(( r + 2 ) \mathrm { cm }\), width \(( r + 1 ) \mathrm { cm }\) and height \(r \mathrm {~cm}\), as shown in Figure 2.
In a set of \(n\) such blocks, the first block has a height of 1 cm , the second block has a height of 2 cm , the third block has a height of 3 cm and so on.
- Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that the total volume, \(V\), of all \(n\) blocks in the set is given by
$$V = \frac { n } { 4 } ( n + 1 ) ( n + 2 ) ( n + 3 ) \quad n \geqslant 1$$
Given that the total volume of all \(n\) blocks is
$$\left( n ^ { 4 } + 6 n ^ { 3 } - 11710 \right) \mathrm { cm } ^ { 3 }$$
- determine how many blocks make up the set.