Use iteration to guess an explicit formula for the sequence...
Materials for Reference:
Use iteration to guess an explicit formula for the sequence... Materials for Reference: Homework Problems Solve the following problems 1. Use iteration to guess an explicit formula for the sequenc...
(1 point) 1 ak= kak-1 for all integers k a) For the following recursive sequence, find the explicit formula through iteration. ao 1 a1 = a3 a4 Explicit formula:
1·2 points Find the first six terms of the following recursively defined sequence: tk(k-1)tk-1 +2tk-2 for k 2 2 1.t1. 2. [3 points] Consider a sequence co, c, C2, . . . defined recursively ck = 3Q-1 + 1 for all k 2 1 and co 2. Use iteration to guess an explicit formula for the sequence 3. [3 points] Use mathematical induction to verify the correctness of the formula you obtained in Problem 2 4. [2 points] A certain...
Consider the sequence defined as a[1] = 2; and a[k] = a[k-1]+2*k-1 for all positive integer k >= 2; . Show that a[n] = 1+sum(2*i-1, i = 1 .. n); . Hint: Start with sum(2*i-1, i = 1 .. n);and use the recursive definition of the sequence.
Problem 3. A ternary string is a sequence of O's, 1's and 2's. Just like a bit string, but with three symbols 0,1 and 2. Let's call a ternary string good provided it never contains a 2 followed immediately by a 0, i.e., does not contain the substring 20. Let Go be the number of good strings of length n. For example, G_1=3, and G. = 8 (since of the 9 ternary strings of length 2, only one is not...
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
(a) Use mathematical induction to prove that for all integers n > 6, 3" <n! Show all your work. (b) Let S be the subset of the set of ordered pairs of integers defined recursively by: Basis Step: (0,0) ES, Recursive Step: If (a, b) ES, then (a +2,5+3) ES and (a +3,+2) ES. Use structural induction to prove that 5 (a + b), whenever (a, b) E S. Show all your work.
(8 marks) Suppose that bo, bi,b2,... is a sequence defined as follows: bo 1, b 2, b2 3, and b bk-1 + 4bk-2 +5bk-3 for all integers k 2 3. Prove by mathematical induction that bn S 3" for all integers n 2 0.
Question 1 10 pts A sequence d1, A2, A3, ... is defined by lettingai = 3 and ak = 70k-1 for all integers k > 2.Show that an = 3 .7h-1 for all integers n > 1. Tips and strategies: • Type many words to explain your reasoning. Use good punctuation. • To access the math editor, just click on the "square root of x" button in your toolkit. The button is indicated in the image below with a red...
11: I can identify the predicate being used in a proof by mathematical induction and use it to set up a framework of assumptions and conclusions for an induction proof. Below are three statements that can be proven by induction. You do not need to prove these statements! For each one clearly state the predicate involved; state what you would need to prove in the base case; clearly state the induction hypothesis in terms of the language of the proposition...
1. Let {an}, be a sequence. Write down the formal definition of the following con- cepts. You have already seen some of these in lecture (a) The sequence is convergent b) The sequence is divergent. (c) The sequence is divergent to oo (d) The sequence is divergent to -oo (e) The sequence is increasing f) The sequence is decreasing (g) The sequence is non-decreasing (h) The sequence isn't decreasing (i) The sequence is bounded above (j) The sequence is not...