2023 CJC Sigma Notation Tutorial Q7

Find the values of \(A\), \(B\) and \(C\) such that for all values of \(r\),

\(1+{{r}^{2}}=A\left( r+2 \right)\left( r+1 \right)+B\left( r+1 \right)+C\).

Hence, prove by the method of differences that \(\sum\limits_{r=1}^{n}{\left[ \left( 1+{{r}^{2}} \right)r! \right]}=n\left( n+1 \right)!\)