6. Let the transition probability matrix of a two-state Markov chain be given, as in Example 4.2, by P =

3 3

3 3

p 1 − p 1 − p p 3

3 3

3 Show by mathematical induction that P(n) =

3 3

3 3

3 1

2 + 1 2 (2p − 1)n 1 2 − 1 2 (2p − 1)n 1

2 − 1 2 (2p − 1)n 1 2 + 1 2 (2p − 1)n 3

3 3

3 3