The Fibonacci numbers are defined as follows: Fo = F; = 1 and for every i2 2, F, = Fi+ F;-2. We thus get the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... Prove by induction that
Q: Use mathematical induction to prove that the statements are true for every positive integer (a)…
A:
Q: Prove that these four statements about the integer n are equivalent: a) n 2 is odd. b) 1 – n is…
A: Set a new statement which is (v) n is odd (i->v) We will prove its contraposition Let n is even…
Q: Copy the problem and answer it on your answer sheets. 5n(n+1) Prove 5 + 10 + 15 + ·… +n =
A: To prove:- 5+10+15+...+5n = n(n+1)/2 Using Mathematical Induction
Q: A natural number n, real numbers a1, ..., an are given. Get all natural numbers j (2 sjs n-1) for…
A: A natural number n, real numbers a1, ..., an are given. Get all natural numbers j (2 < js n-1)…
Q: Prove or disprove the following there is a positive integer such that n+ 4n + 3 is a prime number.
A: You solution is given below.
Q: Prove that f(x) = x is O(x3).
A: Attached image below:
Q: Use induction to prove that 1 + 5 + ... + (4n - 3) = n(2n-1) for all NATURAL numbers n.
A: Induction method can be used by using two cases. In first case we take the value of n=1 and check…
Q: Show that if n is an integer and n3+ 5 is odd, then n is even using a) a proof by contraposition.…
A: a.) Proof by contraposition: The contraposition of the statement is "If n is odd then n3 +5 is…
Q: 2. Prove by Mathematical Induction that: 2.1 4 +95m, m≥ 3 2.2 1x² 10n, m = n, n ≥ 7
A:
Q: 1. Let x ∈ Z. Use a direct proof to show that if 5x2 + 8 is odd then x is odd. 2. Show by…
A: 1) Given 5x2+8 is odd which implies 5x2 is odd as the sum of any odd and even is odd. so 5x2 is…
Q: Use mathematical induction to prove that the following statement is true for all positives integers…
A: Using mathematical induction we have to prove that the given statement is true for all positive…
Q: Suppose we have the following program that computes the quotient and remainder when dividing a by b:…
A: for the above problem given to calculate the quotient and then the following remaining given below…
Q: Prove that"1+3+5+………..+(2n-1)= n2
A: We will prove this by 2 method.
Q: Prove the statement by Mathematical Induction. For all integers ? ≥ 0, 2^2? − 1 is divisible by 3.
A: STEP 1: Let the statement P(n) is given as: P(n): 22n-1 is divisible by 3, for every natural…
Q: Prove the following by using induction: 1(1!) + 2(2!) + 3(3!) +...... + n(n!) = (n + 1)! - 1, for n…
A:
Q: Proof that for all integers numbers n, if n³ is odd then n is odd, using proof by contradiction.
A: Proof by contradiction: For some statement prove it is valid by using proof by contradiction then…
Q: Give a direct proof of: "If x is an odd integer and y is an Question 4. even integer, then x + y is…
A: I will explain it in details,
Q: Prove that the following is true for all positive integers n by using the Principle of Mathematical…
A: In mathematical induction method, first proves that a statement is true for the initial value. Then…
Q: Assume that for any integer n is greater than or equal to one prove or disprove the following…
A: a)n^2 − n + 1 is O(n) ,here Time complexity will be O(n^2). b)5^n is O(4^n),here Time complexity…
Q: Given f(n) E O(n), prove that f(n) E O(n²). Given f(n) E O(n) and g(n) E O(n²), prove that f(n) g(n)…
A:
Q: Prove the following statement using mathematical induction: For all integers n ≥ 0, 3 | ((n^3) + 8n)
A: This question comes from Discrete Mathematics which is a paper of Computer Science. Let's discuss it…
Q: a) i) Give an inductive formula for the sum of the first n odd numbers: 1+3 +5 + .. + 2n -1 Show…
A: As per guidelines, Im supposed to answer the first question, so i request you repost the other…
Q: Suppose we have the following program that computes the quotient and remainder when dividing a by b:…
A: r = a q = 0 while r>=b: r = r-b q+=1
Q: Prove or disprove. show your work. (a) for any integers n a and m: if both n and m are odd, then n-…
A:
Q: Prove truth of formula below using mathematical induction. k(3k +7) 5+8+11+...+(5+3(k–1)) = 2
A: I have answered in this question step 2.
Q: 1 + 3 + 5 + ....... + (2n-1) = n^2 P(k+1) Prove the induction
A: Induction Proof: It is used to prove the particular sequence is true. There are three steps in this…
Q: 12. Show that, if c is a positive real number, then g(n) = 1+c+c² + +c" is: с (a) (e") if c> 1. (b)…
A:
Q: Given that x and y are integers such that 0 < x < y < 9 and that the integer 77265x597y is divisible…
A: Data Given:- x and y are integers 0 < x < y < 9 77265x597y is divisible by 12
Q: State and prove the base case State the inductive hypothesis Outline how the rest of the proof…
A: Lets see the solution in the next steps
Q: Use the Principle of Mathematical Induction to verify that 2 divides n2 + n for all positive…
A: Given: Use the Principle of Mathematical Induction to verify that 2 divides n2 + n for all…
Q: Prove that if n is an integer, then ⌊n/2⌋ = n/2 if n is even and (n − 1)/2 if n is odd.
A: Actually, the answer has given below;
Q: Prove the following equality by induction 12 +22 +32 +...+n2 = (n(n+1)(2n+1))/6
A: Induction: is a mathematical method to prove given formula it is carried out in two steps: step1)…
Q: The given hurricane data are F[0]=4 km/hour and F[1]=6 km/hour. Define what is the value of the 5th…
A: The question has been answered in step2
Q: Carry out all the steps in the the division algorithm for 124 and 22. This means show how to find q…
A: Required:
Q: Discrete Mathematics! Solve the following problems on a piece of clean paper. Show your Solution and…
A: 76 number which is divisible by 3 but not divisible by 4
Q: Prove truth of formula below using mathematica induction. k(3k +7) 5+8+11+...+(5+3(k –1))
A:
Q: Fibonacci principle states that: If we let Xn be the nth integer of the sequence, then the next…
A:
Q: (c) Theorem: If n is an odd integer, then 4 divides n-1. n = 2k + 1 n² – 1= (2k + 1)² – 1 = (4k² +…
A: 1)where n is an odd number that is given so, as per info, we can represent an odd number in the form…
Q: 1. Prove or disprove. (4i-2)=2n² for all natural numbers n z 1. If true, prove using induction. If…
A: i=1 to n ∑(4i-2)=2n^2 now substitute n to n+1 now, i=1 to n+1∑(4i-2)=2(n+1)^2…
Q: ,A+(N-1)·K) and compute the product of all elements For example, if N=3 and K=2, then Walter can…
A: Given:
Q: 1. Prove using induction that n is O(2n).
A: According to the information given:- We have to prove using induction that n is O(2n).
Q: 1. Prove or disprove. integers, then a = b. if alb and bla, where a and b are 2. Prove or disprove.…
A: Dear student, these are multiple questions. As per guidelines , I can do only first three question.…
Q: If a,b,c, and d are consecutive integers, then the sum a+b+c+d is even. Show a direct proof.
A: mathematics proof down below.
Q: proof that if x is an integer and x3 + 11 is odd, then x is even using a proof by contradiction.
A: Given : proof that if x is an integer and x3 + 11 is odd, then x is even using a proof by…
Q: Given the problem of basic multiplication: Input: Two n-digit nonnegative integers, x and y. Output:…
A: O(n)
Q: 7. Show that if n is an integer and n³ + 5 is odd, then n is even using proof by contraposition.
A:
Q: Use Mathematical Induction to prove that: 3 + 7 + 11 + . . . + 4n - 1 = n (2n + 1)
A: To Do: Use Mathematical Induction to prove that: 3 + 7 + 11 + . . . + 4n - 1 = n (2n + 1)
Q: Question 6. x + 2 is odd." Give an indirect proof of: "If x is an odd integer, then
A: To Prove: if x is odd integer than x+2 is odd
Q: Prove the following by proof by cases : min(a, min(b, c)) = min(min(a, b), c) for real numbers a, b,…
A: Here we can have three cases possible Case 1: if a is the smallest real number then clearly a…
Q: Let T(1) = 1 and T(n) = T(n/2) + n², na power of 2. A) Apply mathematical induction to show that…
A: Given that, T(1)=1 T(n)=T(n/2)+n2 n is a power of 2
Step by step
Solved in 2 steps with 2 images
- Correct answer will be upvoted else downvoted. Computer science. stage is a succession of n integers from 1 to n, in which every one of the numbers happen precisely once. For instance, [1], [3,5,2,1,4], [1,3,2] are stages, and [2,3,2], [4,3,1], [0] are not. Polycarp was given four integers n, l, r (1≤l≤r≤n) and s (1≤s≤n(n+1)2) and requested to find a stage p of numbers from 1 to n that fulfills the accompanying condition: s=pl+pl+1+… +pr. For instance, for n=5, l=3, r=5, and s=8, the accompanying stages are reasonable (not all choices are recorded): p=[3,4,5,2,1]; p=[5,2,4,3,1]; p=[5,2,1,3,4]. However, for instance, there is no change reasonable for the condition above for n=4, l=1, r=1, and s=5. Help Polycarp, for the given n, l, r, and s, find a stage of numbers from 1 to n that fits the condition above. In case there are a few appropriate changes, print any of them. Input The primary line contains a solitary integer t (1≤t≤500). Then, at that point,…Informal Proofs Use strong induction to show that every positive integer, n, can be written as a sum of distinct powers of two: 20, 21, 22, 23, ...:1 = 20, 2 = 21, 3 = 20 + 21, ....Show that if f(x) is O(x), then f(x) is O(x2).
- Write an algorithm, called Decomposition_Powers_Three, which produces thedecomposition of each integer using powers of 3, namely 1, 3, 9, 27, and 81, and the +and – operators. Each power of 3 should appear at most once in the decomposition.Examples: 1 = 1 2 = 3 – 1 3 = 3 4 = 3 + 1 7 = 9 – 3 + 1 14 = 27 – 9 – 3 – 1 43 = 81 – 27 – 9 – 3 + 1 121 = 81 + 27 + 9 + 3 + 1Show that the following statements are equivalent, where n is an integer greater than or equal to 2. Feel free to consider any of the following pairs. 1. “n is even” and “n – 1 is odd”2. “n is even” and “n2 is even”3. “n – 1 is odd” and “n2 is even” with Step By step explanation pleaseCorrect answer will be upvoted else downvoted. Computer science. Allow us to signify by d(n) the amount of all divisors of the number n, for example d(n)=∑k|nk. For instance, d(1)=1, d(4)=1+2+4=7, d(6)=1+2+3+6=12. For a given number c, track down the base n to such an extent that d(n)=c. Input The principal line contains one integer t (1≤t≤104). Then, at that point, t experiments follow. Each experiment is characterized by one integer c (1≤c≤107). Output For each experiment, output: "- 1" in case there is no such n that d(n)=c; n, in any case.
- One of the famous proofs of modern mathematics is Georg Cantor’s demonstration that the set ofrational numbers is enumerable. The proof works by using an explicit enumeration of rational numbersas shown in the diagram below.1/1 1/2 1/3 1/4 1/5 . . .2/1 2/2 2/3 2/43/1 3/2 3/34/1 4/25/1In the above diagram, the first term is 1/1, the second term is 1/2, the third term is 2/1, the fourthterm is 3/1, the fifth term is 2/2, and so on.Input and OutputYou are to write a program that will read a list of numbers in the range from 1 to 107 and will printfor each number the corresponding term in Cantor’s enumeration as given below. No blank line shouldappear after the last number.The input list contains a single number per line and will be terminated by end-of-file.Sample input3147Sample outputTERM 3 IS 2/1TERM 14 IS 2/4TERM 7 IS 1/4 bring a brief description (pseudocode) programming language c++Modeling the spread of a virus like COVID-19 using recursion. Let N = total population (assumed constant, disregarding deaths, births, immigration, and emigration). S n = number who are susceptible to the disease at time n (n is in weeks). I n = number who are infected (and contagious) at time n. R n = number who are recovered (and not contagiuous) at time n. The total population is divided between these three groups: N = S n + I n + R n There are several hidden assumptions here that may or may not apply to COVID-19, such as a recovered person is assumed to not be able to get the disease a second time, at least within the time window being examined. On week 0 (the start), you assume a certain small number of people have the infection (just to get things going). Everyone else is initially susceptible, and no one is recovered. There are two constants of interest: Let period = time period that it takes for an infected person to recover (recover meaning they become not infectious to…Show that for f(n) = 2n2 and g(n) = 20n + 3n2 , f(n) is θ(g(n)). How many ways can it be shown? Also Show that for g(n) = 10n2and f(n) = n! + 3 , f(n) is Ω(g(n)). How many ways can it be shown? Discuss with the instructor.
- Prove by Induction that for all integers n ≥ 1, n < n2 + 1 .Yes this problem is silly, but still do it by induction! Prove by Induction that for all integers n ≥ 3, 2n < n2 .Prove the following statement by contraposition.For every integer x, if 5x2 – 2x + 1 is even, then x is odd.What is wrong with the following “proof” that an odd number minus an even number is always 1? Let x be odd and y be even. Then x = 2m + 1, and y = 2m, where m is an integer, and x − y = 2m + 1 − 2m = 1.