![1. Consider the sequence defined recursively by ao = ], Ant1 = V4 an – An, n > 1. (a) Compute ai, a2, and a3. (b) For f(x) =](http://img.homeworklib.com/questions/1801e1f0-fdd7-11eb-933c-1b65fb3a06af.png?x-oss-process=image/resize,w_560)
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1. Consider the sequence defined recursively by ao = ], Ant1 = V4 an – An,...
13. Consider the sequence of numbers ao, ai, a2, a3, given by ao-2, ai-3, and for...
13. Consider the sequence of numbers ao, ai, a2, a3, given by ao-2, ai-3, and for any positive integer k 2, a3ak 2ak-1. (a) Evaluate a2,a3, a4,as. Show your work. (b) Prove that for all positive integers n, an 2 +1
Consider the sequence {an} defined recursively as: a0 = a1 = a2 = 1, an =...
Consider the sequence {an} defined recursively as: a0 = a1 = a2 = 1, an = an−1+an−2+an−3 for any integer n ≥ 3. (a) Find the values of a3, a4, a5, a6. (b) Use strong induction to prove an ≤ 3n−2 for any integer n ≥ 3. Clearly indicate what is the base step and inductive step, and indicate what is the inductive hypothesis in your proof.
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the...
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1 STO X STO 3-X ENTER, ENTER ENTER ENTER. 4-x Apparently the sequence diverges / converges to b. State the MONOTONE CONVERGENCE THEOREM: c. Use induction to show that (x) is decreasing for all n when x, 3 d. Use induction to show that (x.) is bounded below by 0 when x,- 3. e. Conclude from (b-d): d. To...
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F...
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F for all n E N and with Fi = F2= 1. to find a recursive relation for the sequence of ratios (a) Use the recursive relation for (F) Fn+ Fn an Hint: Divide by Fn+1 N (b) Show by induction that an 1 for all n (c) Given that the limit l = lim,0 an exists (so you do not need to prove that...
An array A[1,2,... ,n is unimodal if its consists of an increasing sequence followed by sequence...
An array A[1,2,... ,n is unimodal if its consists of an increasing sequence followed by sequence a decreasing sequence. More precisely, there exists an index k є {1,2,… ,n} such that there exists an indes . AlE]< Ali1 for all 1 i< k, and Ai]Ali 1 for all k< i< n A1,2,..,n] in O(logn) time the loop invariant (s) that your algorithm maintains and show why they lead to the correctness Give an algorithm to compute the maximum element of...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may eq...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao =...
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...
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn +...
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn + (-1)-1 for n 2 1. Then X n is a decreasing sequence. an increasing sequence. a Cauchy sequence either increasing or decreasing. QUESTION 12 Check if the following statement is true or false: COS n The sequence is divergent. True False
3. A sequence is a map a N°R, typically written (an) = (ao, a1, a2, a3, a4,) As an example, the sequence (an) = 1/(...
3. A sequence is a map a N°R, typically written (an) = (ao, a1, a2, a3, a4,) As an example, the sequence (an) = 1/(n2 +1) begins (1, 1/2, 1/5, 1/10, 1/17,..) Here is a useful fact relating sequences and continuity: A function f(x) is continuous at x c if and only if for every sequence (an) that converges to c, written anc, then f(x,) f(c). Alternatively, if you and f(yn)L" with L' L", then f is not continuous at...
Given the sequence an defined recursively as follows: an 3an-1+2 for n 2 1 Al Terms of a Sequence...
Given the sequence an defined recursively as follows: an 3an-1+2 for n 2 1 Al Terms of a Sequence (5 marks) Calculate ai , аг, аз, а4, а5 Keep your intermediate answers as you will need them in the next question. A2 Iteration (5 marks) Using iteration, solve the recurrence relation when n21 (i.e. find an analytic formula for an). Simplify your answer as much as possible, showing your work and quoting any formula or rule that you use. In...
13. Consider the sequence of numbers ao, ai, a2, a3, given by ao-2, ai-3, and for any positive integer k 2, a3ak 2ak-1. (a) Evaluate a2,a3, a4,as. Show your work. (b) Prove that for all positive integers n, an 2 +1
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1 STO X STO 3-X ENTER, ENTER ENTER ENTER. 4-x Apparently the sequence diverges / converges to b. State the MONOTONE CONVERGENCE THEOREM: c. Use induction to show that (x) is decreasing for all n when x, 3 d. Use induction to show that (x.) is bounded below by 0 when x,- 3. e. Conclude from (b-d): d. To...
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F for all n E N and with Fi = F2= 1. to find a recursive relation for the sequence of ratios (a) Use the recursive relation for (F) Fn+ Fn an Hint: Divide by Fn+1 N (b) Show by induction that an 1 for all n (c) Given that the limit l = lim,0 an exists (so you do not need to prove that...
An array A[1,2,... ,n is unimodal if its consists of an increasing sequence followed by sequence a decreasing sequence. More precisely, there exists an index k є {1,2,… ,n} such that there exists an indes . AlE]< Ali1 for all 1 i< k, and Ai]Ali 1 for all k< i< n A1,2,..,n] in O(logn) time the loop invariant (s) that your algorithm maintains and show why they lead to the correctness Give an algorithm to compute the maximum element of...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
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...
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn + (-1)-1 for n 2 1. Then X n is a decreasing sequence. an increasing sequence. a Cauchy sequence either increasing or decreasing. QUESTION 12 Check if the following statement is true or false: COS n The sequence is divergent. True False
3. A sequence is a map a N°R, typically written (an) = (ao, a1, a2, a3, a4,) As an example, the sequence (an) = 1/(n2 +1) begins (1, 1/2, 1/5, 1/10, 1/17,..) Here is a useful fact relating sequences and continuity: A function f(x) is continuous at x c if and only if for every sequence (an) that converges to c, written anc, then f(x,) f(c). Alternatively, if you and f(yn)L" with L' L", then f is not continuous at...
Given the sequence an defined recursively as follows: an 3an-1+2 for n 2 1 Al Terms of a Sequence (5 marks) Calculate ai , аг, аз, а4, а5 Keep your intermediate answers as you will need them in the next question. A2 Iteration (5 marks) Using iteration, solve the recurrence relation when n21 (i.e. find an analytic formula for an). Simplify your answer as much as possible, showing your work and quoting any formula or rule that you use. In...
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