Problem
ALG-B1-M05-P011 Divisibility of Terms
#11
★★★☆☆ Level 3 of 5
The sequence is defined by \(a_1=4\), \(a_{n+1}=5a_n+4\). Prove that \(a_n\) is divisible by \(4\) for all \(n\).
Use induction.
Base case is clear: \(4\mid a_1\). If \(4\mid a_n\), then \(5a_n+4\) is also divisible by \(4\). Hence \(4\mid a_{n+1}\).
Simple induction on a recurrence.