Challenging +1.2 This is a two-part proof question requiring (1) algebraic manipulation of factorial expressions to verify Pascal's identity, and (2) a structured induction proof of the binomial theorem. While it requires formal proof technique and careful bookkeeping with the Pascal's identity result, both parts follow standard templates that Further Maths students practice extensively. The induction step is mechanical once Pascal's identity is established, making this moderately above average but not requiring novel insight.
4 Using factorials, show that \(\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }\).
Hence prove by mathematical induction that
$$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$
for every positive integer \(n\).
4 Using factorials, show that $\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }$.
Hence prove by mathematical induction that
$$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$
for every positive integer $n$.
\hfill \mbox{\textit{CAIE FP1 2016 Q4 [6]}}