n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

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Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

Q: Problem 1: (a) Use mathematical induction to prove that 157 12"+1 + 132n-1 for all n 2 1. Perform…

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Q: Find the smallest positive integer n such that 5 (3² +2²) (3ª +2¹) (3³ +2³)... (3² +22") > 9256

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Q: 8. Prove that n!+1 and (n+1)!+1 are integer n. relatively prime for every positive

A: Given, and are two numbers and to prove that they are relatively prime.

Q: An integer n > 2 is composite if and only if n doe

A: Given Statement: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1.…

Q: 5. By Principle of Mathematical Induction, prove 2+ 22 + . 2" = 2n*1-2 for all n>1 %3D ...

A: Hi, thank you for the question. Since you have posted two questions, as per our policy we will…

Q: For a natural number n, we define n! = 1·2·3· ... · (n − 1) · n. For example: 2! = 1 2,3! = 1 2 3 =…

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Q: 10. Use induction to show that 1x2 + 2x3 + + n(n+1) = " for all n € N.

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Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: (d) 21 1 4"+1 + 52n-1.

A: Hey, since there are multiple questions posted, we will answer question d as you specified. If you…

Q: prove that for all integers m, n & d if d/m & d/n then d/(2m+3n)

A: We know that if a and b are integers such that a/b then b is a multiple of a. i.e. b=ka where k∈ Z{…

Q: 2. If P (n) : "2.4z + 3""" is divisible by k for all ne N" is true, then the value of k is

A: The solution is given below:

Q: 5. (20) Prove by induction that for positive integers n, 4|(5"+1 + 32n-1).

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Q: Prove by induction that, for every n\geq2 n+1 (1-2)(1-¹)(1-(1-1/2) = ² 2n

A: To Prove: We have to prove by induction that for every n≥2, we have:…

Q: or, Recurrence Formulae : ¨ (2n + 1) xP₂ = (n + 1) Pn+j+nP,-1 (n+1) Pn+1 = (2n + 1) xP₁ − nP,. -

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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: (2n- 2. Give a combinatorial argument to show that k- ways the number of ways to select a committee…

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

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Q: The factorial of n is the product of all positive integers less than or equal to n. It is…

A: Note:Hey there! Thank you for the question! We have solved this problem using trial-and-error…

Q: 4. Prove by induction that 3" ≥ 2" +n for all n ≥ 1.

A: We have to show that assuming the statement is true for n = k we have to prove the statement is true…

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: (b) 3|2" + (–1)"+1 for all n > 1.

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Q: 12 Let u₁ = 1, ₂ = 1 and un + 2 = Un + 1 + un n ≥ 1. Then use mathematical induction to show that 1…

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Q: 6. Prove that n, n+2, n +4 are all primes if and only if n = 3.

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Q: Exercise 6.4 (1) For which natural numbers n is the number 3" + 1 divisible by 10? Find the…

A: (1) We have to find the natural numbers for which the number 3n+1 is divisible by 10. (2) We have to…

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A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: If n is a positive integer, then prove that 2n(2n-1) 2n(2n-1)(2n-2) 2! 3! 1-2n+- + ...

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Q: (c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n

A: We know that,

Q: Let's practice a proof by induction! 1 1 %3D 1.2 2.3 n (n+ 1) n+1 for any integer n > 1.

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Q: The multiplication rule says that if an operation can be performed in k steps and, for each i with 1…

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Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…