Correct option is (C) \(\frac{n(n+1)(n+2)(n+3)}{4}\)
\(S_n=\) 1.2.3 + 2.3.4 + 3.4.5 + …….. + upto n terms
= 1.2.3 + 2.3.4 + 3.4.5 + …….. + n (n+1) (n+2)
\(=\sum n(n+1)(n+2)\)
\(=\sum n(n^2+3n+2)\)
\(=\sum n^3+3n^2+2n\)
\(=\sum n^3+3\sum n^2+2\sum n\)
\(=\left(\frac{n(n+1)}2\right)^2+3\left(\frac{n(n+1)(2n+1)}6\right)+\frac{2n(n+1)}2\)
\(=n(n+1)\left(\frac{n(n+1)}4+\frac{2n+1}2+1\right)\)
\(=\frac{n(n+1)}4(n^2+n+4n+2+4)\)
\(=\frac{n(n+1)}4(n^2+5n+6)\)
\(=\frac{n(n+1)(n+2)(n+3)}{4}\)
