Evaluate $$\frac{1\times 2^2 + 2\times 3^2 + \ldots + n\times (n+1)^2}{1\times 2 + 2^2\times 3 + \ldots + n^2\times (n+1)}.$$

Write the numerator term as $$n(n+1)^2=n^3+2n^2+n,$$ and the denominator term as $$n^2(n+1)=n^3+n^2.$$Hence the given expression becomes $$\frac{\sum_{k=1}^n (k^3+2k^2+k)}{\sum_{k=1}^n (k^3+k^2)}.$$Now use $$\sum k=\frac{n(n+1)}{2},\quad \sum k^2=\frac{n(n+1)(2n+1)}{6},\quad \sum k^3=\frac{n^2(n+1)^2}{4}.$$Substituting and simplifying carefully gives $$\frac{3n+5}{3n+1}.$$Option A is the reciprocal of the correct result, a common error if numerator and denominator expansions are interchanged.