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}}