Prove that the fraction 7n+1/ 20n+3  is irreducible for every natural number .

Q: Prove that there exists exactly one prime of the form n^3 -1. (When we say that a number has "the…

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Q: Use mathematical induction to prove that (3n + 7n − 2) is divisible by 4 for all integers n ≥ 1.

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Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

A: Either n is a prime or it is composite; in the former case, there is nothing more to prove. If n is…

Q: prove via Mathematical Induction that 3n +1 is divisible by 2 for all n>=1

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Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: Use induction to prove that for all n>0 in natural numbers , 1/1+1/4+1/9+...+1/n^2 is less than or…

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Q: If there are (2n+1) term in AP.then prove that the ratio of the sum of odd terms and the sum of even…

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Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: Prove that every positive integer n>1 has a prime factorization.

A: Prime factorization is a way of expressing a number as a product of its prime factors.

Q: 1- gn+1 Σ %3D :- 1-τ i=0

A: Solution

Q: 2. Prove that the number of irreducible fractions not greater than 1 and with denominator equal to n…

A: For the solution of the problem follow the next step.

Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

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Q: 7n+1 20n+3 is irreducible for every natural number n. Prove that the fraction

A: Solution is given below:

Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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Q: Use induction to prove that f (n) = 8" – 1 is divisible by 7 for all natural numbers n. Hint: For…

A: Proof by induction involves three step. Take the statement to be proved as P(n). First step is to…

Q: 11n+1 +122n-1 is divisible by 133 for all natural numbers

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Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

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Q: Prove or disprove: 3n = ω(2^n ).

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Q: 21 divides 4n+1 + 52n-1

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Q: Use induction to prove that 5^(2n) − 4^n is divisible by 7 for all natural numbers n.

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Q: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Use mathematical induction to prove that 2n+2 + 32n+1 is divisible by 7 for all integers n 2 0.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Use induction to prove 23n-3n is divisible by 5 (for all natural numbers n > or equal to 1)

A: To prove 23n – 3n is divisible by 5 (for all-natural numbers n ≥ 1) using induction method.

Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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Q: Prove that for any integer n if 3n+ 2 is even, then n is also even.

A: As per Bartleby answering guidelines if multiple questions are posted in a single picture then an…

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: 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: 1. Prove that, for any positive integer n," n´+2n is divisible by 3 " using the method of…

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

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

Q: prove that it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

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Q: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

Q: Use generalized induction to prove -    1 + 2n < 2n2 for all integers n greater than or equal to…

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Q: 4. Prove by induction that 21 divides 4"+1 + 52n-1 for all positive integers n.

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

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 simple continued fraction of a real number is unique i.e. if [a0,a1,...] = x =…

A: To Prove: The simple continued fraction representation of a real number is unique, i.e. if…

Q: infinitely many primes of the form

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Q: By Principle of Mathematical Induction prove 5n – 1 is divisible by 2 for all n >= 1

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Q: Prove that there exists no rational number x such that x2 = p, where p is a prime number (A prime…

A: We need to prove that there exists no rational number x such that , x2 = p  ,  where p is prime…

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: Exercise 1. Show that there are infinitely many primes of the form 4n +3. Hint: you might find it…

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Q: Prove by induction that for positive integers n: 1! x3! x5! x...(2n (2n-1):2 (n!)"

A: For n=1 Left hand side = 1!=1 Right hand side  =n!n=1!1=1 now 1≥1 The result is true for n=1 Assume…

Q: Prove that there are infinitely many primes of the form p = 4n + 3.

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Q: Prove that 2n^3 - 2n is divisible by 6 for every n>=2, by mathematical induction

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Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

Q: Use induction to prove: For all integers n ≥1, the integer  5n –1 is divisible by 4.

A: For n=1, 51-1=4 which is divisible by 4. So, the result is true for n=1 Let the result is true for…

Q: Prove that there exists exactly one prime of the form n^3 −1. First express the statement…

A: Number

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.