(a) The n-th Mersenne number can be defined recursively by Mo = 0 andn-1M₁ = n + ΣM₁i=0for n ≥ 1.i. Compute M, for 0 ≤ n ≤ 5.ii. Find a recursive definition for Mn that has order 1, that is, the recursiveformula for M₁, only involves Mn-1 (and possibly some constants).iii. Find an explicit formula for Mn.

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Answered: (a) The n-th Mersenne number can be…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3.4. A sequence s,, S2, S3, ... is defined recursively as follows: Sk = 5Sk-1 + (Sk-2)² for all integers k 2 3 S1 = 4 S2 = 8 Use (strong) mathematical induction to prove that s, is divisible by 4 for all integers n > 1.
Given the recursive formula, ƒ (n) = f (n − 1) + ƒ (n − 2), n ≥ 3, and f(1) = f (2) = 1. What is f (8)?
Please explain the steps
4) Consider the following recursive formula an = an-1- V3, where n2 2and the first term is a = 12. Which of the following could be the explicit equation? A an = 12 – V3-n, where n>0 B a, = 12 – V3 -n, where n20 c an = 12 - V3 -n, where n 21 D an = 12 - V3- (n – 1), where n20 E) n = 12 - V3 - (n – 1), where n 21
(b) Modular arithmetic Let x, y, N € N be n-bit integers such that N > 2. • If x < N, the value x" mod N can be computed using the following recursive formula: x" mod N = 1 X gcd (x, y) = The number of bit operations when using this method is O(.....). • If x ≥ y, the value gcd(x, y) can be computed using the following recursive formula: , if y = 0 , if y ≥ 1 -₁ , if y = 0 , if y = 1 , if y ≥ 2 even , if y ≥ 2 odd The number of bit operations when using this method is O(.....).
This is a multi-part question. e an answer is su you will be unable to return to this For each of these functions, find the least integer n such that f(x) is O(x). of 2 NO k ces If 8 (x²+z²+1) (24+1) then the least integer nis
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For the following, write its recursive definition:
Example 1.5.4, the decimal 0.101001000100001000001..., is related to an 1844 construction by Joseph Liouville, who proved that the number is transcendental. The decimal expansion of Liouville's number can be thought of as a sequence of blocks, each block consisting of a digit 1 followed by a string of zeros. Liouville's example differs from Example 1.5.4 because now the strings of zeros increase in length extremely fast. Let j be the number of times the digit 1 appears in the first 50 decimal places of the number in Example 1.5.4, and let k be the number of times the digit 1 appears in the first 50 decimal places of Liouville's number. Then j - k = 1 1 10n! 3 5
(a) Give a recursive definition of the following two structures ) the sequence a,n = 3" for every positive integer n. (i) The set A consisting of positive integer powers of 3. That is A = (3"|n e Z*) (b) Describe the similarities and differences between (i) the different structures you are given (e.g. the sequence a, and the set A) and (i) the recursive definitions you produce.
The average depreciation rate of a new boat is approximately 8% per year. If a new boat is purchased at a price of $75,000, which model is a recursive formula representing the value of the boat n years after it was purchased?->a 1) a = 75,000(0.08)” a = 75,000(1,08)" 11 n 2) a= 75,000 a = 75,000 a₁ = (0.92)" a₁ = 0.92(a,, -₁) 3) 4) 6 Savannah just got contact lenses. Her doctor said she can wear them 2 hours the first day, and can then increase the length of time by 30 minutes each day. If this pattern continues, which formula would not be appropriate to determine the length of time in
6. Discover the recursive formula for Hn. Find Hn for n=1,2,3...6. Based on these values you will be able tosee the recursive pattern. proving the formula is optional for extra credit. Consider all subsets of N={1, 2, 3,..., n). How many of these subsets, SC N, satisfy the property that (i) Vx (x ES → x +1 ¢ S) ? Denote the number of subsets satisfying this property (i) by Hn. (a) Find the first 6 values of Hn . I have calculated n=1 and n=2, so you need to do n=3, 4, 5, 6 (b) Find and prove the recursive formula for the sequence Hn . Example: Let n=1. Then N={1} and the subsets satisfying property (i) are {1} and O so H1=2. Let n=2. Then N={1,2} and the subsets satisfying property (i) are {1}, {2}, and Ø, i.e. H3=3. Note: {1,2} is also subset of {1,2} but since for x=1, x+1=2 and both 1 and 2 are contained in {1,2} therefore the set {1,2} fails to satisfy property (i). 7. Prove by induction that the number of subsets for a set with n elements is 2^n.

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