Standard +0.8 This is a 'suggest and prove' induction question requiring students to spot the pattern S_n = (n+1)! - 1 from numerical examples, then prove it. While the induction proof itself is straightforward once the formula is conjectured, the pattern recognition step and working with factorials adds moderate difficulty beyond standard induction exercises. This is typical Further Maths content but not exceptionally challenging.
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 [7]}}