1. Consider a sequence {an} defined by recurrence: a1 =1, and an = 2an-1+5 for n 2. Prove using induction that an = 3 2n - 5 for any n21. 2 subsets A and

1. Consider the sequence 3, а2 -9 , an 7ап-1 — 10аm-2, for n > 3. a1 2n+2 – 5" for all n > 1. Use strong induction to prove that an

Give a recurrence relation that would fit the sequence a1, az, A3, A4, A5, .= 1,2, 5, 26, 677 ...

Find a recurrence relation for the number an of n-digit ternary sequences, i.e. sequences consisting of {0,1,2} only, such that the subsequence 012 does not occur. In particular, you must write down a sufficient number of initial conditions, and then derive your recursive formula with the pattern explained. Hint: Consider using complements in your solution.

Find a recurrence relation for the number of sequences of yellow and purple stones of length N, where each sequence has the property that no two adjacent stones are purple. HINT: Treat sequences whose final stone is yellow separately from sequences whose final stone is purple.