Standard +0.8 This is a two-stage induction problem requiring algebraic manipulation of the recurrence relation before proving divisibility. Students must simplify a_{n+1} - a_{n} to find a useful recurrence, then apply induction with modular arithmetic. The combination of establishing the recurrence and proving divisibility by 24 elevates this above standard single-step induction proofs, but it remains a structured Further Maths exercise with clear guidance.
3 It is given that, for \(n = 0,1,2,3 , \ldots\),
$$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$
Simplify \(a _ { n + 1 } - a _ { n }\), and hence prove by induction that \(a _ { n }\) is divisible by 24 for all \(n \geqslant 0\).
3 It is given that, for $n = 0,1,2,3 , \ldots$,
$$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$
Simplify $a _ { n + 1 } - a _ { n }$, and hence prove by induction that $a _ { n }$ is divisible by 24 for all $n \geqslant 0$.
\hfill \mbox{\textit{CAIE FP1 2002 Q3 [6]}}