If $$S = 4^2 + 2 \cdot 5^2 + 3 \cdot 6^2 + \ldots + 25 \cdot 28^2$$, then the value of $$\frac{S}{325}$$ is equal to

The general term can be expressed as

$$n(n+3)^2 = n^3 + 6n^2 + 9n$$ for $$n$$ from 1 to 25. 

$$\sum n(n+3)^2 = \sum n^3 + 6\sum n^2 + 9\sum n$$

$$\sum n^3 = \dfrac{n^2(n+1)^2}{4} =   \dfrac{25^2(25+1)^2}{4}= 105625$$

$$\sum n^2 =\dfrac{n(n+1)(2n+1)}{6} =\dfrac{25(25+1)(50+1)}{6} = 5525$$ 

$$\sum n =\dfrac{n(n+1)}{2}=\dfrac{25(26+1)}{2}=   325$$, 

We get $$S = 105625 + 6(5525) + 9(325) = 141700$$. 

Hence $$\dfrac{S}{325} = 436$$.