1. Apply the AM-GM inequality to prove that for any n = N, (n + 1)" ≥ 2nn!.

Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: 5. Prove that, for n> 1, n

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

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Q: Is it possible to find a number  n  such that is ∫0∞  xn dxconvergent?

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Q: Show that (n) ‡ 14 for all n ≥ 1.

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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 sum of the first n natural numbers, S = 1 + 2 + 3 + first n natural numbers is 171, determine…

A: Consider the given information:

Q: Prove: n! > 2"- for any positive integer n that is greater than 2.

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Q: . Let n be a positive integer. Prove that 1 · ( n/ 1)  + 2 · ( n/ 2)  + . . . + n ·  (n/ n) …

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Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: Solve for the moment generating function of X if for all natural number n, X ∼ U {1,2,…,n}.

A: Given that, X ∼ U {1,2,…,n} , we need to solve for the moment generating function of X if for all…

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

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Q: 2. (Arithmetic Mean-Geometric Mean Inequality) Let n E N and {r;}"-1 be nonnegative numbers, prove…

A: Use mathematical induction and check if the statement is true for n=1. The left hand side of the…

Q: Prove Γ (n) = (n-1)! where n ∈ ℕ

A: Concept: The proof for this gamma integral will be done using mathematical induction

Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

Q: (b) Prove that E-o(-1)* )= 0 for all positive integers m.

A: We have to prove ∑k=0n-1kkn=0 for all positive integer n

Q: 4. Prove that for every n > 1,3 divides n³ – n.

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

Q: Show that. 1 (n + 1)(n + 2) n=1

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Q: Prove the following equation for every n>=2 by mathematical induction.…

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Q: 12. Let the "Fibonacci-2" numbers gn be defined as follows: 91 = 2, g2 = 2, and gn+2 = (gn+1)( In)…

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Q: 4. Prove that 3"< n! if n is an integer greater than 6.

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: 4. (P11, Рage 203 (Bernoulli's Inequality) Show that for a natural number n and a nonnegative number…

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Q: What is the smallest integer n such that d(n) = 8? and d(n) = 10?

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Q: (a) Show that if n € N, then <¹dr. n+1

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Q: B={n ∣ n ∈ N and 214 < n ≤ 134} n(B)=

A: We are given the description of a set B such that : B={n ∣ n ∈ N and 214 &lt; n ≤ 134} We are asked…

Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…