Show that
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r$$
can be expressed in the form
$$k n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$
where \(k\) is a rational number and \(a , b\) and \(c\) are integers.
Show that there is exactly one positive integer \(n\) for which
$$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r = 8 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$