(a) Show that n(n² – 1)(n+2) is divisible by 4 for all integer n. (b) Prove by contradiction : if n – m is odd , then n² – m² is odd .

Q: Let n ∈ Z. Prove that if n is divisible by 3, then n = 6k or n = 6k + 3 for some integer k.

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Q: Show, using the principle of mathematical induction, that 2 + 6 + 10 +… + (4n-2) is equal to 2n ^ 2…

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Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Use mathematical induction to prove that (n³ + 3n² + 2n) is divisible by 3 for all n > 1.

A: See the attachment

Q: Find all integers n>1 such that 1n + 2n +... + (n-1)n is divisible by n.

A: In the given question n>1, there arises two cases as follows- Case 1- When n is odd that is…

Q: - 1 Use the factorization A = PDP to compute A", where k represents an arbitrary integer. I а 6b-а)…

A:

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

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Q: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

Q: (b)(i) Use Mathematical Induction to prove that for all positive integers n 21 1 1. 1 一 -十 +... 1x3…

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

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Q: 17. Which is a step in showing that n³ + 2n is divisible by 3 is true by mathematical induction?…

A: On solving this we will get

Q: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

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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: 2. Use the binomial theorem to show that for all integers n 2 1, 3" = 6) + 2 (4) + 2ª (") + 2" (")

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Q: Which is a step in showing that n° + 2n is divisible by 3 is true by mathematical induction? O Show…

A: The solution is given below

Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: 2 is a factor of n2 + 5n.Use mathematical induction to prove that the statement is true for every…

A: To explain the given problem

Q: 2 is a factor of n2 + 3n.Use mathematical induction to prove that the statement is true for every…

A: To prove : n2+3n is divisible by 2 or 2 is a factor of n2+3n for every positive integer n. Proof :…

Q: Prove that for any n E N, nº + 2n is divisible by 3.

A:

Q: 1 + 3 + 5 +. . . + (2n - 1) = n2.Use mathematical induction to prove that the statement is true for…

A: To prove: 1+3+5+...+2n-1=n2 by mathematical induction. Formula used: a+b2=a2+2ab+b2 Proof: Assume Pn…

Q: Give a combinatorial proof of the identity: -1 k(²) = n(n=¹) k-1.

A: Use the following combinatorial formulae, to obtain the result. nr=n!r!n-r!n!=nn-1!

Q: true or false In the domain of Gaussian integers Z[i], the element 17 is Irreducible

A:

Q: SKETCHPAD Question 11 Use mathematical induction to prove that, for n2 1, n³ - n is divisible by 3.

A:

Q: r(r+1) n+1 8. Prove by mathematical induction that 5"-4n - 1 is divisible by 16, for all positive…

A: As per our guidelines we are supposed to solve only one answer . Please repost another question as…

Q: 1 + 2n < n° for all integers n² 2

A:

Q: . Prove, using mathematical induction, that 13n – 8n is divisible by 5, for n ≥ 1.

A: Let us consider the statement is  P(n) : 13n - 8n is divisible by 5. For n = 1, 13n - 8n = 5 is…

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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Q: #2. The binomial theorem states that for any real numbers a and b, In (a + b)n = =Σ" ("a²-kbk for…

A:

Q: 4 + 8 + 12 + . . . + 4n = 2n(n + 1).Use mathematical induction to prove that the statement is true…

A: To prove: 4+8+12+...+4n=2nn+1. It is proved by mathematical induction. For n=1 4=211+1=4 Here, the…

Q: 3. Use the binomial theorem to prove that 2n 2n |22n–2k = (2.5)2". k k=0

A:

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Prove that2.7n + 3.5n – 5 is divisible by 24, for all n ∈ N.

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Q: [-4 0-4] 3. Let A = 0 5 0 (a) Prove by mathematical induction that A" =5A for all positive odd…

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Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

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Q: Prove that n3+(n+1)3+(n+2)3 is divisible by 9 for all non-negative integers  n.

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Q: Find (a) an algebraic proof, and (b) a combinatorial proof for the following: Σ |2*3n-k k 4*. k k=0…

A: We have to find an algebraic proof, and combinatorial proof for ∑k=0nnk2k3n-k=∑k=0nnk4k  ⋯1.

Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: Find all integers n> 1 such that 1" + 2"+... +(n – 1)" is divisible by n.

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Q: Prove that for all n E N, 7" +2 is divisible by 3.

A: We will use mathematical induction to prove the result in question 4.

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

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Q: 2. a) Use the principle of mathematical induction to prove the statement: 6 7" – 2· 3" is divisible…

A:

Q: P

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Q: a. Use the Binomial Theorem to write out the complete expansion of (x³ + y²)$ With all coefficients…

A: Binomial expansion

Q: Prove by contradiction, with example problem: "For any integer n, if n2 is divisible by 4, then n is…

A:

Q: Prove, using mathematical induction, that 8n−3n is divisible by 5 for all integers n≥0.

A:

Q: 3. Prove that if p is a prime >3 then the numerator of 1 + 1/2+1/3+...+V/(p-1) is divisible by…

A: Given: Let us consider the given, r=1+12+13+...+1p-1

Q: Prove that n(n + 1)(n + 2) is always divisible by 6

A: