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

Q: Prove that 1– 2+2² – 2° + -.-+(-1)"2"=2**1 for all nonnegative integers n.

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Q: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

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

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: Prove: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

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Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: Prove that 2n > n for all positive integers n.

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

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Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n > 0.

A: Let a be a non zero real number , where m and n are integers…

Q: Abstract Algebra

A: To find non-negative representations under the given conditions

Q: 9. Prove that V2 is an irrational number.

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Q: 60. Prove that for any positive integer n, for every real number x. %3D

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: n + 2020 n + 2021 and LVn +1]

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Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: #3. Let n be an integer with nz1 and Let b, bz . bn be ... positive real numbers, not all egual,…

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Q: For all integers n>0, prove: 2n < (n+2)!

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Q: Provide either a proof or a counterexample for  For all positive real numbers x, x^2 - x ≥ 0

A: According to the given information, it is required to prove the given equality or give a counter…

Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Use the definitions of even and odd to prove: For all integers n, n2+3n is even.

A: We have to prove, For all integers n, n2+3n is even.   We prove this by using Principal of…

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: Compute the supremum and infimum of the set A = {n/(2n+1) : n ∈ ℕ}

A: We have to find the supremum and infimum of the following set A =n(2n+1):n∈ℕ

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

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

A: ANSWER:

Q: 2. Let x be irrational real number. Prove that 2x+3 is irrational

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Q: Prove that 11 is the only prime number of the form 4n2 - 25.

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Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: Let's take an integer n to be odd if there is some integer k such that n = 2k – 1. Prove that the…

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Q: prove that there is no positive integer n such that n^2+n# = 100

A: n^2+n^3 = 100 We can evaluate this till some positive integers

Q: prove that (n(nt)* s an integer V impers

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Q: coth- x = In () Q1/ Prove that

A: To prove coth-1x=(1/2)ln[(x+1)/(x-1)]

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: Show that |x^n| = |x| ^n for every positive integer n and every real number x.

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Q: Prove that 3" >n² for all positive integers

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Q: Prove by contradiction that if a sum of three real numbers is less than 30, th at least one of the…

A: We have to prove by contradiction that if a sum of three real number is less than 30, then at least…

Q: Let a relatively prime positive integers. If there exist integers s and t with as + ht t - 11 then

A: if gcd (a,b) =d then there exists integers m and n such that d = a m + b  n  and d is the smallest…

Q: 6. Prove that for all natural numbers n, 2n +n <3".

A: Let Pn:2n+n≤3n The statement is true for n=1 because 21+1=3=31. Now, suppose that for n=k, the…

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: 2. Show that for any real number x, if x + 2 > 10, then x? – 14 > 5x. Use direct proof.

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Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…