If P(n): 2n < n! , nEN then P(n) is true for all n 2... None n23 n22 n24

Q: 2. (Proof by Contradiction) For every n E Z, 4 t (n² – 3).

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Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

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Q: 0.3.14: Prove 13 +23 + - +n³ = ()) (mla n(n+1) for all n E N.

A: Mathematical induction

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

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Q: ^n for all n>=6

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Q: The statement 2n< n! is valid for For only n=4 All positive integers All positive integers n 2 4 All…

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Q: (a) Use mathematical induction to prove the following inequalities: (i) 2n +1<n² for n 2 3 (ii) n² +…

A: We are going to prove the following inequalities by using mathematical induction .…

Q: Prove that n/n2 +1 is greater than (n+1)/ (n+1)2 + 1

A: If An and Bn be two term. To show that An is greater than Bn it is sufficient to show that…

Q: *3. Prove that 1² +2² + ... · +n² = n(n+1)(2n+1) for all n € N.

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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Q: b) Prove the following by mathematical induction: n(n+1)(n+2) 1.2+2.3+3.4+..+n(n+1) : 3 for all…

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Q: 2 3 n a. If n E N, then + + + •+ 3! 4! (n+1)! (Hint: 3! (reads 3 factorial) = 3 x 2 x 1. n! = nx…

A: Induction theory: Let the given statement is P(n). Step-1: We need to check P(n) is true for n=1.(or…

Q: f(n)={1 if n is odd; n if n is even  Prove or disprove. f(n) is O(n).

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Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: सिद्ध है कि n! > 3 ^ n सभी पूर्णांकों के लिए n> = 7।

A: हमें दी गई समस्या को हल करना है:

Q: Prove the given statement: For 2 ≤ n ≤ 4, n2 ≥ 2n.

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Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: 1 For all integers n21:+ 1.2 2-3 3. n(n +1) n+1

A: Calculate the left hand side of the given equation. 11·2=12 Calculate the right hand side of the…

Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: . Prove that 2 +4+6+•…+ 2n = n(n + 1) for all n > 1.

A: According to the given information it is required to prove

Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we…

A: Principal of mathematical induction

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

A: here is the proof

Q: For all integers n> 1, 2+4 + 6+ . ..+ 2n = n² +n.

A: Here we are going to solve only the first question and for the remaining question please post the…

Q: P(n): 2"+1 > 2n + 1

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Q: (-1)*-1 '(nIn n)? n=1

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Q: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

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Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: Q4) Prove for any natural number n 20 Li 1+2+2+. +2" = 2-1 ミ (n+1) .... (11) 1+2+3+ + n = 2

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

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Q: (3a) Let n E Z. Prove that if 3 ł n, then 3|(n² – 1)

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Q: 3. Use the permutation and combination formulas to prove the following. (a). For all integers n 2 3,…

A: Given n≥3, n€ Z (integer ) We have to prove that P(n, 3)+2P(n, 2)=nP(n, 2)

Q: Assume that p > 1. For all n>1 prove that p – 1 (n + 1)P <n'-P – (n+ 1)'-P.

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Q: 2. Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 + 5n, express an in terms of n.

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Q: Q5/ prove by mathematical induction, that for all n 21 1+5+9++(4n- 3) = n(2n-1)

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

Q: 6. Prove o (k2) = k · Þ(k) for all k E N.

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Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: (b). For all positive integers n 2 2. (") = 2 (5) + n².

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Q: 04 Prove that m (n-k) = (n+1)- ("-m)n 2m n2 m 2

A: Note: Cr+1n+1=Crn+Crn-1+…+Crn-m+Crn-m-1+…+Crr→1 From identity 1, Crn-m=Crn-m-1+Crn-m-2+…+Crr Now,…

Q: Use the permutation and combination formulas to prove the following (a). For all integers n > k > 2,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: only Q2(8) 2. Show that following statements are correct: į. 4n+100=0(n) iii. n³ ± O(n²) v. n! =…

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Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: For all n E N,n > 1. there exists a prime p such that p|n.

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Q: Prove that for any natural n the following equality holds (n – 1)n(n+ 1) 1.2+2.3+...+(n – 1) · n =

A: We have to prove that for any natural n,            1⋅2+2⋅3+......+n−1⋅n=n−1nn+13. We will prove…

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…