Show that \( \frac{1 \times 2^{2}+2 \times 3^{2}+\ldots+n \times(n+...
Show that \( \frac{1 \times 2^{2}+2 \times 3^{2}+\ldots+n \times(n+1)^{2}}{1^{2} \times 2+2^{2} \times 3+\ldots+n^{2} \times(n+1)}=\frac{3 n+5}{3 n+1} \).
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