Question 1: 2 a,b: *s E la:/² E 16:12.† i=1 i=1 i=1 For the above inequality ai, bị, 1 < i

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: Prove: | a | - | b | &lt;= | a + b |

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Q: Prove by direct proof that any integer n ≥ 3 can be part of a Déjà vu Pair n = [11 · · · 1]b for…

A: Given: Any integer n≥3 can be part of a Déjà vu Pair n=11...1b for some base b≥2.   To prove: The…

Q: Give a direct proof of the theorem “The sum of any two odd integers is even”.

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Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

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Q: Prove that for all positive integers n, 1 3 2n – 1 1 X...X 4 V3n 2n HINT: You will find it difficult…

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Q: Prove. A number is triangular if and only if it is of the form (n(n+1))/2 for some nonnegative…

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Q: If A = {r|- 23 < x < 3how many negative integers does Acontain? Note that: 'I' means 'Such that'

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Q: Prove by mathematical induction

A: The principle of mathematical induction is a technique or method that is used to prove the statement…

Q: Use a direct proof. Let "a", "b", and "c" be integers, where "a" is not equal to 0. “if a|b and b|c,…

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Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: The following “Theorem” is obviously not true. Explain what’s wrong with the proof. Theorem: 1 is…

A: Given Data: Given Theorem is : 1 is the largest natural number. This theorem is not true. We have to…

Q: ) Prove every integer n >= 7, n! > 3n

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Q: Prove by strong induction that root of 2 is irrational.

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Q: 7. Ug = kuz-1¬- Ug–2, for every integer k > 3 uj = 1, u2 = 1

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Q: Is there a real number that is bigger than all integers? If so what is the number and if not what…

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Q: Prove there are no natural numbers n and p with n ≥ 2 and p² + 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 using mathematical induction that the following inequality holds for n ≥ 3: 4n-1 > n²

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Q: Write down the supremum of the set {7+ 7/n2 − 5/m | n,m ∈ N}. prove ur answer from the definition.

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Q: 2. Prove that n + 100 > 3" if n is a positive integer with 1 < n < 3. (Hint: Proof by cases)

A: Concept:

Q: Prove that cuberoot(2) is an irrational number

A: We can prove that the cube root of 2 is an irrational number using proof by contradiction.Let us…

Q: Prove: For all integers m and n, if m +n is even, then m and n are both even or m and n are both…

A: What to prove: For all integers m and n, if m+n is even, then either m and n are both even or m and…

Q: 4. Prove that (a + b) ≤a+b≤ a + b for any real numbers a and b. (Hint: Use cases when a, b ≥ 0; a, b…

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

Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: 2. Prove using induction that the following inequality holds for n2 0: (1+)"21+ n

A: First we show this is true for n=0,1 . Then we assume this is true for n=m , and we shall show this…

Q: Prove that if sum of two numbers is irrational then at least one of them is irrational. Use indirect…

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Q: Use a combination of techniques such as proof by contrapositive and proof by cases to prove the…

A: The contrapositive statement is : "If x is an integer, then (x³ - x) is even" (1) Assumption : we…

Q: only d)

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Q: d/ Prove that the following numbers are real (for any 3 享

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Q: If x and y are two positive real numbers, then there exists a positive integer n such that ny>x

A: Proof: Suppose there does not exist any positive integer n such that ny&gt;x then ny≤x   ∀n∈ℕ

Q: Prove the inequality x² + x² + ... + x² n IV 2 2+...+ ²₂ ) ² n

A: Explanation of the answer is as follows

Q: Show how to adapt the proof that v3 is irrational (from the video) to prove that v15 is irrational.…

A: 3 is irrigational number: Let 3 is rational number then 3=pq,where p, q are both integers and q≠0…

Q: Consider the following theorem: "The sum of two even integers is even." Rewrite the theorem as a…

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Q: a. Show that for any real number x, if x> 1 then | x4 |≤| 23x4 + 8x² + 4x |.

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