(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: For a non negative whole number n, let the function fn be given by (picture). where n!= 1*2...n,…

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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: 2. The Fibonacci numbers F(0), F(1), F(2),… are defined as follows: F(0) := 0 F(1) := 1 F(n) := F(n…

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Q: 1 1.3 1 Question 2. + + 3.5 1 5.7 + ... + 1 (2n-1) (2n + 1) for n ≥ 1

A: As per the question we are given the following series : 1/(1×3) + 1/(3×5) + 1/(5×7) + ... +…

Q: 1. Use mathematical induction or strong mathematical induction to prove the given statement. 1 (a).…

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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: Prove C(n,0)=C(n,n)=1 for all whole numbers n

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Q: Show that for all n ≥ 1, the n ₁²>(n+¹) " +1\ in

A: We have to show that for all n⩾1 , the inequality                  ∏j=1njj⩾n+12n(n+1)2Since…

Q: The factorial of a positive integer n, denoted n!, is equal to the product of all integers between 1…

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Q: Prove the following by using induction: 1(1!) + 2(2!) + 3(3!) +...... + n(n!) = (n + 1)! − 1, for n…

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Q: (a) Let n EN. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when divided…

A: PART A As it is given that 2n+1, 2n+3, 2n+5 are consecutive odd integers so they are pairwise…

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: Show that n 1 i2 i=1 <2 - n

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Q: (a) Prove that for n ≥ 2, 3 > 2n + n². (b) For any n ≥ 1, the factorial function, denoted by n!, is…

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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: . Use Mathematical Induction to show that for all n > 1. 71 i=1 F2i = F2n11-1

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Q: Prove that 1¹x 2² x 3³ x...xn" < n(n+1) 2 (2n+1) 3

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Q: For all integers n>0, prove: 2n < (n+2)!

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Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

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Q: Use the Geometric Sum identity to prove that, if r 70, then (1 – x)3 1 =1+ (1 - x) + (1 – x)² + |

A: The objectives to use geometry sum identity to prove the following expression.

Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

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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

Q: Which of the following statement is always false? (a) For all integer x, [x] = [−x] (b) For all odd…

A: We have to check which of the given option is always false. a Given that  for all integers x,  x=−x.…

Q: 1 = (₁ + - +/- ) ² (b) Prove that n < 1+ for all n € N.

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Q: Show that for n ≥ 3, ô(n) is even.

A: We have from the definition of Euler ϕ function: ϕ(1)=1 and ϕ(2)=1 Now Let n ≥ 3

Q: Prove:  C(n,k) = C(n-1,k-1)+C(n-1,k) for natural numbers k with 0<k<n.

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Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…

Q: -3an The solution of the recursion relation ant1= n > 0, with ao = 1, (n+1)? " is an O None of these…

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Q: Show that...

A: We have to show that: n1=nn-1 for n≥1. We know the formula, nr=Crn=n !r ! n-r ! Applying above…

Q: Use Bernoulli's inequality to prove that n + √)" = 1+ > √n for all n EN.

A: From Bernoulli inequality we know that  if x&gt;-1 then : 1+xn≥1+nx for n≤0 or n≥1. Therefore for…

Q: 1=1,f2 =1 and fn+2 =fn +fn+1 for all n≥1.    Prove that for all natural numbers n≥1: f1 f2 +f2 f3…

A: As the domain for the result is natural numbers, n≥1, we can use principle of mathematical induction…