Let ao, a1, a2, ... be a sequence of real numbers defined by the recurrence relation given below: do := 4 for each integer n > 1, an = 3an-1+4 Using a Proof by Induction prove that, for all integers n > 0, the nth term of the above sequence equals 5 – 5-". That is, prove the following statement P(n) holds true for all integers n > 0: P(n): а, 3D 5 — 5-т Important: Include all relevant details in your proof. If variables appear in your proof, clearly indicate what they represent. Clearly indicate your Induction Hypothesis and indicate exactly when it is used in the proof of your Induction Step.

Determine if the sequence {b„} is a solution to the recurrence relation an = 2an-1 + 3an-2, Vn > 2. %3D a) bn = 3n+1, Vn > 0 b) bn = n+1, Vn > 0 %3D %3D

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

A sequence a,,a, ,az,..., satisfies the recurrence relation a, = 3a +10a2, with initial conditions a, = 5 and a, = 8. Find an explicit formula for the sequence. Use this formula to calculate a,0. %3D %3D

2. Let an be the number of ways to write the numbers 1, 2, 3, ..., n in order such that no number i appears immediately before i + 1 (this is a question which appeared on the exam, where an explicit formula was requested!). Explain why an will satisfy the recurrence an = (n-1) an-1+ (n − 2)an-2 for every n ≥ 2.

Write a recurrence relation and initial conditions for the number Sn of sequences of Nickles, dimes and quarters that can be inserted into a vending machine to purchase a soft drink costing 5n cents. How many sequences are there for a drink costing 50 cents?

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.

Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation fk = f, f, = 1 + 2* for each integer k 2 2 - 1 satisfies the following formula. f. = 2" +1 - 3 for every integer n 2 1 Proof (by mathematical induction): Suppose f,, fa, fa,... is a sequence that satisfies the recurrence relation f, = f, -1+ 2* for each integer k 2 2, with initial condition f, = 1 We need to show that when the sequence f,, f, f, ... is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above. So let the property P(n) be the equation f. = 2" +- - 3. We will show that P(n) is true for every integer n 2 1. Show that P(1) is true: The left-hand side of P(1) is which equals The right-hand side of P(1) is Since the left-hand and right-hand sides equal each other, P(1) is true. Show that for each integerk2 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that P(k) is true. In other…