Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation k = -1 + 2* for each integer k 2 2 f, = 1 satisfies the following formula. = 2n +1- 3 for every integer n 2 1 Proof (by mathematical induction): Suppose f,, f, 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, fa, ... 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 +1- 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 words, suppose that f, = , where f,, f2, f3 ... is a sequence defined by the recurrence relation f, = f,- + 2* for each integer k 2 2, with initial condition f, = 1. [This is the inductive hypothesis.) We must show that P(k + 1) is true. In other words, we must show that f, 41 = Now the left-hand side of P(k + 1) is k +1 = f, + 2* + 1 by definition of f, fe, fi, + 2k +1 by inductive hypothesis v %3D = 2 - by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete. [Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

Let an be the number of strings (i.e. sequences of letters, non-sense words) of length n that can be formed using the four letters Q,X,T and Z. (a)Find an explicit formula for an (b)Find a recurrence relation that is satisfied by the numbers an. (c) Find a rational generating function (ratio of two polynomials) that is a generating function for the numbers an.

Consider the task of arranging red, white, and blue flags on an n-foot flagpole with no gaps, where red flags are 4- feet high, white flags are 2-feet high, and blue flags are 1-foot high. Which of the following recurrence relations can be used to determine the number distinct ways x(n) of arranging red, white, and blue flags on the n-foot flagpole for n=1,2,3, ... if the appropriate initial conditions are known? O x(n) = x(n-3) + x(n-2) + x(n-1), n = 5,6,7, .. O x(n) = 4x(n-4) + 2x(n-2) + x(n-1), n 5,6,7, . O x(n) = x(n-4) + x(n-2) + x(n-1), n = 5,6,7, .. O x(n+4) = x(n) + x(n-1) + x(n-2), n = 1,2,3, .. • Previous Next Simpfun

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

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.

Discrete Mathematics a) Find a recurrence relation for the number of sequences of 1s, and 2s whose terms sum to n.   b) repeat this problem with the added condition that no 2 can be followed by a 2.

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.