1. Prove that for all numbers of k, (5*2^(2k+1) - 1)/3 is an odd number.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Prove that 7^n − 1 is a multiple of 6 for all n ∈ N. ( N is Natural Number)

A: Advance maths

Q: NOT ALL THE NUMBERS ARE ON HERE!!

A: Solution-: Given data: 18,80,83,295,332,489,503,513,530,544,570,606,793,840,912,980 We want to find…

Q: how that in the domain of integers of the form a+bsqrt(-17) the factorization…

A: The  objective is to show that 13 is irreducible. The norm of 13 is 13=132=169 So if 13 is reducible…

Q: integer greater than 1 can be written in the form n = p₁ p2p with a_{i} > 0 and p, are distinct…

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Q: Prove that for all integers, n , n^3-n is divisible by 3.

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Q: Write a formal induction proof (Base Case and Inductive Step) Prove by induction that…

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Q: Find the smallest distinct postive numbers that provide a counterexample to show the statement is…

A: Let the odd numbers be 1 and 3 and the even number be 2, According to the question, The sum of any…

Q: Use the Second Principle of Finite Induction to establish that for all n is greatre than or equal to…

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Q: I was asked to prove that 5^(2n+1) + 4^(2n+1) is divisible by 9 for all n that are positive…

A: SOLUTION-

Q: Prove or disprove the following statement: There are prime numbers a, b, c such that a^2+ b^2= c^2.

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Q: How can I prove by induction that 2^n + 1 is divisible by three for all odd integers

A: To apply mathematical induction to show that 2^n+1 is divisible by 3 if n is odd

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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Q: सिद्ध है कि n! > 3 ^ n सभी पूर्णांकों के लिए n> = 7।

A: हमें दी गई समस्या को हल करना है:

Q: Prove that for all natural numbers k, a (5*2^(2k+1) - 1)/3 + a*4^k is of the form 5 + 6a.

A: The question looks incorrect and cannot be proved. This is because of the following.…

Q: How can you prove or disprove: 3^n ≥ 2^n + 5n for all integers n ≥ 3?

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Q: Prove that for every natural number n, 1+3+5+...+(2n-1)=n^2

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Q: K = (perfect squares of integers} under addition? Is it closed or not

A: K is the perfect square of integerNow we find whether it is closed or not.

Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

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Q: bers {1, 3, 6, …}

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Q: If n is a positive integer, then prove that 2n(2n-1) 2n(2n-1)(2n-2) 2! 3! 1-2n+- + ...

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Q: Find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime. Justify that your…

A: We have to find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime.

Q: Prove that for all natural numbers k, a (4^k - 1)/3 + 2*a*4^k is of the form 1 + 6a

A: We show that the question doesnt hold for a=1 and k=1

Q: If O is a natural number then O^2 + 2O + 1 is odd.

A: If O is a natural number then we need to check whether  O^2 + 2O + 1 is odd. The result is given…

Q: Use mathematical induction to prove that 2^(n+2)+3^(2n+1) is divisible by 7 for all integers n≥0.

A: We have to prove that  2n+2+32n+1 is divided by 7 by using principle of mathematical induction for…

Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: expand each expression using the Binomial Theorem

A: Given: 19.(x-2)6

Q: Exercise 2.8. Let a, b be real numbers and let a ^= (₁ 1) A and B -(1) What is AB ? Compute A² and…

A: We use method of induction to prove A^m .

Q: Which of the following gives the Binet form of the nth Fibonacci number? O Fn = Fn = Fn = Fn =…

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Q: For n is greater than or equal to  1, use mathematical induction to establish each of the following…

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Q: Given an odd integer a, establish that a^(2) + (a+2)^(2) + (a+4)^(2) +1 is divisiible by 12 using…

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Q: write a program to evaluate even numbers for [1,10],and [11,20]respectively in matlab

A: Even numbers for [1,10] is given below-

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

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Q: Prove that for all natural numbers k, (4^k - 1)/3 is an odd number.

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