CAIE FP1 2011 June — Question 4

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJune
TopicProof by induction

4 It is given that \(\mathrm { f } ( n ) = 3 ^ { 3 n } + 6 ^ { n - 1 }\).
  1. Show that \(\mathrm { f } ( n + 1 ) + \mathrm { f } ( n ) = 28 \left( 3 ^ { 3 n } \right) + 7 \left( 6 ^ { n - 1 } \right)\).
  2. Hence, or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 7 for every positive integer \(n\).