3 It is given that \(u _ { r } = r \times r !\) for \(r = 1,2,3 , \ldots\). Let \(S _ { n } = u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots + u _ { n }\). Write down the values of $$2 ! - S _ { 1 } , \quad 3 ! - S _ { 2 } , \quad 4 ! - S _ { 3 } , \quad 5 ! - S _ { 4 }$$ Conjecture a formula for \(S _ { n }\). Prove, by mathematical induction, a formula for \(S _ { n }\), for all positive integers \(n\).

3 It is given that $u _ { r } = r \times r !$ for $r = 1,2,3 , \ldots$. Let $S _ { n } = u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots + u _ { n }$. Write down the values of

$$2 ! - S _ { 1 } , \quad 3 ! - S _ { 2 } , \quad 4 ! - S _ { 3 } , \quad 5 ! - S _ { 4 }$$

Conjecture a formula for $S _ { n }$.

Prove, by mathematical induction, a formula for $S _ { n }$, for all positive integers $n$.

\hfill \mbox{\textit{CAIE FP1 2014 Q3}}