Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

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: 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: 16- Prove that for every positive integer n and all real number r except 1: -1 r -1 i=0 17- Show…

A: We need to prove the following given condition in question 16 and 17

Q: Prove that for all positive integers k and n, with k < n, n-1) k-1/ k 1 k ... k k- 1 k- 1

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

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Q: 7. Use induction to prove Bernoulli's inequality: If 1 + x>0, then (1+ x)" al+ nx for all natural…

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Q: Prove by the principle of mathematical induction that 1 x 1!+2 x2! + 3 x 3! +…+ n x n!= (n+1)!—1…

A: to prove the result 1×1!+2×2!+3×3!+...+n×n!=n+1!-1 for all natural numbers n , proceed as follows…

Q: Let n≥1, x be a real number, and x ≥ −1. Prove the following statement using mathematical induction.…

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Q: 1. Prove that nº + n is even for every integer n.

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Q: 9. Using a proof by induction, prove that for every natural number n the quantity 11n -6 is…

A: The statement is not true. Tale n=4

Q: 1- gn+1 Σ %3D :- 1-τ i=0

A: Solution

Q: Prove that for all integers a, b, and c if a divides b then for all naturla numbers n a^n divides…

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Q: Prove that 2" >n For all nuatural numbers

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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 for every natural number n 1° + 2° + 3* + ... +n° = -n°(n+ 1)². %3D

A: Method used : Mathematical Induction

Q: Use mathematical induction to prove that for every positive integer n, the sum of the first n…

A: 1+2+3+---+n=nn+12 When n=1, ⇒1=11+12⇒1=1 Which is true. Therefore given statement is true for n=1.…

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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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 for all positive integers k and n, with k < n, (:) - () - (^) n k 1 k n 1 k k – 1 k – 1 k…

A: 1+kk-1+K+1k-1+   ... ...+n-1k-1 add and subtract 1 and write 1 as a combination of k objects taking…

Q: 1) Let x be a positive real number. Show by induction on n that (1+x)" > 1+ nx for all integers n >…

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Q: Prove by Induction that for every natural number n, 3"-1 is a multiple of 2?

A: Use principle of induction to prove this.

Q: Prove by mathematical induction that for every positive integer n, 1 1 1 1 П 3i 3і Зі + 1 (Зп + 1)!…

A: Since you have posted multiple questions and we can answer one question at a time. So, answered…

Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

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Q: Prove that (5n)!÷40^n×n! is natural number for every natural number n​

A: To Prove that (5n)!÷40^n×n! is natural number for every natural number n​

Q: Let n > 1 be an integer. Prove that ℤ/n has exactly n elements.

A: Prove that ℤ/n has exactly n elements. Let n&gt;1 be an integer. That is, n=2, 3, 4, ....∈ℤ ℤ is an…

Q: (1 + x)" > 1 + nx

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Q: Suppose that a particular real number has the property that (x + (1/x)) is an integer. Use (strong)…

A: Given that x+1x is an integer. We have to prove that xn+1xn is an integer for all natural number n…

Q: Prove that a positive integer n is p

A: Topic :- Number system

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

A: ANSWER:

Q: Prove that For every real number x > 0, there is a natural number M such that for all natural…

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

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

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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: 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 for every n in the set of natural numbers, N:

A: The partial fraction is the process of splitting the terms in the denominator into different…

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: Let n be an odd positive integer. Find all perfect numbers of the form nn +1.

A: n=1,3,5,7,9,11,13,...nn+1=11+1,33+1,55+1,77+1,99+1,... Sequence is: 2, 28, 3126, 823544, 387420490,…

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: Prove that for any integer n, if n³ is odd then n is odd.

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Q: Prove by induction that the sum of the first n odd numbers is n². In other words, prove E1(2i – 1) =…

A: To prove by induction that the sum of the first natural numbers is n^2.

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: Define a sequence of real numbers as follows: Let x1 = 1, and then for every n ∈ ℕ, define xn+1 =…

A: Let x1=1 and then for every n∈ℕ, define xn+1=xn2+1

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: Prove by induction on n that, for all positive integers n: Σο (i(i!)) = (n + 1)! – 1

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

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…