(b) For any n ≥ 1, the factorial function, denoted by n!, is the product of all the positive integers through n: n! = 1.2.3.. ... (n − 1) · n Prove that for n ≥ 4, n! > 2n.

Q: Let = n n i=1 x₁ . Then for constants a and b, n Σ a(x₁ - ī) b² =? (6²5m)

A: The question is about parametric inference.Given :Where,X1, X2, ...., Xn are independent random…

Q: Verify that the following inequality is true for all integers n > 1 1 1.3.5. .... (2n 1) - 2n…

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Q: For all n ≥ 1, let an = (a) Find a1, a2, A3, A4. (b) Computea. i=1 (c) Computea. i=1 (n + 1)² n!…

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Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

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Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: (1) How much is 0! ? (by convention on). 0!

A: (1) Given

Q: (c) Prove that for n ≥ 1, n [27/7/2≤2- j² 1 n

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Q: () is even. 3. Prove that for all positive integers n, the number

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Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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Q: For integers n>r> 0, use the definition of the factorial function to prove that (n − 1)! (n − 1)! n!…

A: By the definition of factorial function, the factorial of a non-negative integer n, denoted by n!,…

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: use gauss's trick to determine R(n)=2+3+4+...+(n-1)+n+(n+1)

A: It is known that sum of first n numbers is n(n+1)2  that is  1+2+3+----+n=n(n+1)2 Gauss trick method…

Q: Let A = 21. (a) Compute A², A³, and A4. 02 (b) Find a formula for A" for any integer n ≥ 1?

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Q: Prove that if |x-1| < 1, then x² - 2x + 5| < 8.

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Q: Show that the generating function for the number of compositions of n where each part is either 2 or…

A: To find the generating function for the number of compositions of n where each part is either 2 or…

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: x₁ + x₂ + +x. Prove that -x₁)(1+x₂)... (1+x) ≤1+S+ ... S² S³ + 2! 3! +...+ uS - n!

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Q: True or False: The summation 18+28+...+n8 is a polynomial of n of degree 8.

A: Given that the summation 18+28+...+n8. The objective is to check the statement " the summation…

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

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Q: Prove that n Sk k=1 = n(n + 1)(2n + 1) 6 for all n € N.

A: To prove:We will use mathematical induction to prove this.Let