Question 2) Prove that if n is odd then n divides 2-1.

Q: Question: 10 Induction to prove Big-O Prove by induction that n! is O(n") for n21

A: We will prove it using mathematical induction.

Q: (5n)! 40"-n! is natural number for every natural number n. Prove that

A:

Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: Show that if n is odd and 3 n, then n2 = 1 (mod 24).

A: The solution is below.

Q: If there are (2n+1) term in AP.then prove that the ratio of the sum of odd terms and the sum of even…

A:

Q: Let n and k be natural numbers with n >k. how many subsets of {1,2, ..., 2n} have exactly k even…

A:

Q: Question 5 Find all integers n >1 such that 1" + 27+...+(n – 1)" is divisible by n. |

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n Proof: Let…

Q: Question 7 For an integer n, if n is even, then n- is even. True False

A:

Q: Question 9 ( What can we do to show that 313 is prime? We can determine whether the smallest prime…

A: Prove that 313 is a prime number.

Q: Prove that the sum of the 1st natural n numbers = n*(n+1)/2

A:

Q: QUESTION 9 Use Mathematical induction to prove that 2 divides n2+5n, n21.

A: Question 9: Let n=1 is true for Let the result be true for n=k,  We need to prove the result is…

Q: -, 1+2+2 '+.+ 2 -1 for all natural numbers n.

A: We will use the principle of mathematical induction to prove the result.

Q: How many four-digit combinations can be formed using digits 0 to 9?

A: Total number of digits n=10 Number of digits for combination r=4

Q: Prove that if n is a multiple of 4, then Fn (Fibonacci n) is a multiple of 3.

A: Fibonacci function is defined by F1=1,F2=1, Fn=Fn-1+Fn-2, n>2 We will prove that F4m is a…

Q: Prove or disprove: for every natural number n,2+5+8+...+(3n-1)=n(3n+1)/2

A:

Q: How many even 4-digit numbers can be formed from the digits 1, 2, 6, 7, and 9 with no repetitions…

A: There is no repetition allowed.. At unit place we have two choices 2 or 6 In 10th place we have…

Q: 11**) (Solution to be discussed in tutorial only) For a polygon to be conver means that all of its…

A:

Q: Question 2. Show that for any integer n, 5|(n5 − n).

A: The given statement is that "for any integer n, 5|n5-n". By Principle of Mathematical Induction,…

Q: QUESTION 4 Find two integers, x, where -3 < x < 12 such that x =4 mod 6 X = and x = YOU MUST SHOW…

A:

Q: Prove N.又

A: Note as per our guideline we are allowed to do 1 question at a time. Please post other questions…

Q: prove that "The Sum of two odd numbers is even: using direct. Note:- 1 want the ansawer on a paper.

A: The sum of two odd numbers is even. Proof :          Properties odd and even integer.          If…

Q: APRL In homework problem 17 in section 3.1 you showed that there are an infinite number of primes…

A: We will use basic knowledge of number theory to answer this question.

Q: Prove that the nth Fibonacci number fn≤αn−1, where α is the golden ratio.

A:

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

A:

Q: What mistakes are found in this proof.

A:

Q: So what's the final value of x1 x2 x3 and x4? indivually

A: Answer :  No value  the system of equations does not converge to any value. so there doesn't exist…

Q: Question 3. Let n > 2 be an integer. Show that n is either a prime or it has а prime faclor < Vп.

A: Question: Let n be an integer such that n≥2. Prove that either the number is a prime number or it…

Q: How many integers from 1 to 1000 are divisible by at least one of  the numbers 2, 3, 5?

A: Considering integers from 1 to 1000, Total integers  = 1000 Let n(A) represent the integers…

Q: - 1, 1, 9; degree: 3 omial with integer coeffi

A: Is an expression consisting of variables (also called indeterminants) and coefficients, that…

Q: Claim: all natural numbers are either even or odd. Proof: By induction. Base case: 0 is even, since…

A: 1)   Yes. The proof of first claim is correct.  Here use 1st principal of mathematical induction.

Q: proximat 5 in [2, 3) 11

A:

Q: Theorem 4.6 (Fundamental Theorem of Arithmetic) Every positive integer n can be written as the…

A:

Q: countable in number, such that

A: Given : W=Iα:α∈A is a collection of open intervals, then there exists a sub-collection , say Ii :…

Q: QUESTION 8 Find the smallest integer greater than 100 that is divisible by 3 but leaves the…

A:

Q: How many different ways could you choose any 4 gumballs from the 25 gumballs in the…

A: We can choose 4 gumballs from the 25 gumballs in 25C4 ways.

Q: Demonstrate by induction if n > 2, then: 1- (4) + G)- ;) n. .3 = 0 * NOTE: consider odd and even…

A:

Q: Question 5: Prove that if nEN is such that 2" - 1 is a prime number, then n must be a prime number…

A:

Q: Prove that 2^? +3^? is a multiple of 5 for all odd ?∈N.

A: We will solve this problem using principle of mathematical induction.

Q: Prove that the simple continued fraction of a real number is unique i.e. if [a0,a1,...] = x =…

A: To Prove: The simple continued fraction representation of a real number is unique, i.e. if…

Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A:

Q: (5n)! 40 n! is natural number for every natural number n. Prove that

A:

Q: Problem 1. Does there exist a natural number n such that 13 (n² + 1)? Prove your assertion.

A:

Q: Question 5 (5n)! is natural number for every natural number n. 40" n! Prove that

A:

Q: Prove that the sum of the first n natural numbers is given by this formula: n(n + 1) 1+2 +3 +. ..…

A: As per our guidelines, we are supposed to answer only one question. Kindly repost other questions as…

Q: Find the sum of all integers from 41 to 83. The sum of all integers from 41 to 83 is

A: Detailed Solution and Explanation is given below....

Q: For all ? ≥ 1, ?3 + 2? is divisible by 3. Prove using Principle of Mathematical Induction.

A: Let P(n) be the given statement i.e.,  P(n): n3 + 2n is divisible by 3 (for all n greater than or…

Q: Question 3. of 24. Show that if n is an odd integer, then n³ - nis multiple

A:

Q: The product of all the natural numbers from 1 to n is written as n! it is read as "n factorial" true…

A:

Q: Theorem 4.6 (Fundamental Theorem of Arithmetic) Every positive integer n can be written as the…

A: Given fundamental theorem of arithmetic: Every positive integer n can be written as the product of…

Q: Prime divisors of two numbers a and b are 2,3 and 5 and product of these numbers a and b is 5400.…

A: The prime factors of the two numbers are 2, 3, and 5 and the product of these two numbers is 5400.