Consider two G. Ps. \( 2,\,2^{2},\,2^{3},\ldots \) and \( 4,\,4^{2},\,4^{3},\ldots \) of \( 60 \) and \( n \) terms respectively. If the geometric mean of all the \( 60+n \) terms is \( (2)^{\dfrac{225}{8}} \), then \( \displaystyle \sum_{k=1}^{n} k\,(n-k) \) is equal to:
