Knowing that for every real numbers a and b, (a + b)² = a² + 2ab + b², one can easily prove that for every integer n, (n + 1)² – n² is odd by means of: a. a direct proof b. a proof by cases c. a proof by induction

Q: 2.) Let 1 > -1 be a real number. Prove that (1+1)" >1+ nx for all integers n > 1.

A: We use principal of mathematical induction (PMI) to prove this statement.

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

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Q: Suppose that the number 'a' has a property that for every natural number n, a ≤ 1/n. Prove that a ≤…

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Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contradiction.

A: Given: 3n + 2 is an even integer Let n is not even, thus n is odd. Therefore, by using the…

Q: 1. Prove that for every real number x, if |x-3| > 3 then x² > 6x. (Hint: Consider two cases)

A: Solution

Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Prove that if n is an integer, then 3n? + n + 14 is even

A: Case I - Let n be an odd integer , then n= 2k+1 where k is an integer. Then 3n2+n+14   =…

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove that for any natural number N, there exists N consecutive integers none of which is a power of…

A: Please see the attachment

Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Prove that for each irrational number r, there exists a unique integer n such that |r – n| < 1/2.

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Q: Prove that the statement is true for every positive integer n.

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

Q: Let a and b be natural numbers, where a,b >= 2, such that gcd(a,b)=1 and ab=(x+y)n for some natural…

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

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Q: Prove by Induction that if                                          a1,a2,a3,.........an belongs to…

A: I have proved for n=1, 2 and by assuming n=k i have proved for n=k+1

Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Show that if n is an integer and n³ + 5 is odd, then n even using a) a proof by contraposition. b) a…

A: Proofs: The proof is a valid argument that establishes the truth of a statement. Types of Proofs:…

Q: Prove that for all integers a, b, and c, if a2 + b2 = c2, then at least one of a and b is even.

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Q: Let a, b, c be distinct positive real numbers and nbe a natural number. If a, b, c are either in…

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Q: Prove that “if n3 + 5 is odd, then n is even” by using the following: a) Direct Proof b) Indirect…

A: If n3 +5 is odd then n is even. Given:  n3 +5 = 2K+1                (K is any integer)       n3  =…

Q: a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

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Q: Q 2: Suppose r is a real number other than 1. Prove using mathematical induction that for every…

A: Proof by induction

Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

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Q: A sequence {am} of real numbers is defined recursively by a1 = 1, a2 = 2 and a, = (i–1)a; for n > 3.…

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Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n. 22+42+62+...+2n2=2n2n+1n+13 We prove…

Q: Prove there are no natural numbers n and p with n > 2 and p² + 1 = 2".

A: ANSWER:

Q: Consider the statement: for all integer k, if k is even then 4k is even. Prove the statement. Which…

A: Given that k is even then we have to prove that 4k is also even. Since, k is even so, we can write…

Q: Prove, using any method, the following statement: if n is an integer, then (3n+1)(5n+4) is an even…

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Q: Prove that if n even, then n is an integer and 3n+2 is even using is a) A proof by contraposition.…

A: Please refer to the image below

Q: Prove that 1 + 3 + • • • + (2n - 1) = n2 for all natural numbers n.

A: Our Aim is to prove that 1 + 3 + .... + (2n - 1) = n2 for all natural numbers n.

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: Prove that every natural number n can be written in exactly one of the following forms: • 3k for…

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Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Use the method of proof by cases to prove that for any integer n the product n(n + 1) is even

A: Given,  for any integer n the product n(n + 1) is even  We have to prove it using the method of…

Q: Prove that the statements here are true for every positive integer n. 2 + 22 + 23 +……++ 2n = 2(2n -…

A: The given statement is: 2+22+23+...+2n=2(2n-1) We need to prove that the given statement is true for…

Q: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

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Q: Let n>=2 and k be any integer, then prove that (n-1)^2 divides (n^k )-1 iff n-1 divides k.

A: See the attachment

Q: Prove that if n is an integer and n³ + 5 is odd, then n is even using a) A proof by contraposition.…

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Q: 2. Show that if a and b are positive integers and alb, then a < b. 3. Show that if a and b are…

A: Basic facts about division will be used in solving this problem.

Q: a) Prove that if n is an even integer, then n2 + 5n+2 is even. (Use a direct proof.) b) Prove that…

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Q: Prove that there is an infinity of natural numbers n such that 2n +1 is divisible by n3

A: Solution of the problem as follows

Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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Q: 3. Show that for any real number x, if x2 is odd, then x is odd.

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Q: 1 Prove that if z is a real number such that z+- is an integer, then for all natural numbers 1 n, z"…

A: Given that, if z is a real number such that: z+1z is an integer,Show that for all n,zn+1zn is an…