Suppose that an is a sequence recursively defined as follows: 5. An = 5. (as) +...
Suppose the language L ? {a, b}? is defined recursively as follows: ? L; for every x ? L, both ax and axb are elements of L. Show that L = L0 , where L0 = {aibj | i ? j }. To show that L ? L 0 you can use structural induction, based on the recursive definition of L. In the other direction, use strong induction on the length of a string in L0. 1.60. Suppose the language...
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.
PROVE BY INDUCTION Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by...
1. Consider the sequence defined recursively by ao = ], Ant1 = V4 an – An, n > 1. (a) Compute ai, a2, and a3. (b) For f(x) = V 4x – x, find all solutions of f(x) = x and list all intervals where: i. f(x) > x ii. f(x) < x iii. f(x) is increasing iv. f(x) is deceasing (c) Using induction, show that an € [0, 1] for all n. (d) Show that an is an increasing...
14. (15 points) Recall that Fibonacci numbers are defined recursively as follows: fnIn-1 +In-2 (for n 2 2), with fo 0, fi-1 Show using induction that fi +f 2.+fn In+2-1. Make sure to indicate whether you are using strong or weak induction, and show all work. Any proof that does not use induction wil ree or no credit.
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...
Suppose that 20, 21, 22, ... is sequence defined as follows. do = 5,21 = 16,0 integers n > 2. Prove that an = 3.2" +2.5" for all integers n > 0. = 7an-1 – 10an-2 for all
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described in this procedure, as follows: (a) Write the linear approximation 1 (x) to the curve at the point (Xn,f(xn). (b) Find where this linear approximation passes through the x-axis by solving L(x)0 for x. xn + 1-1,-I n). is the recursion formula for Newton's Method. : Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described...
(1 point) Find the first six terms of the recursively defined sequence 251/2 n-1 Sn = for n > 1, and s1 = 1. 4. first six terms = (Enter your answer as a comma-separated list.)
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...