Part III: Prove or Disprove the following 8. Vn e Z, if n is prime, then 2" +1 is also a prime number

Q: 2p a) n! = 2 (mod p), where for 0 <k < n. %3D k!(n - k)!

A: To prove the given part (a) 2pp≡2mod p where nk=n!k!n-k! for 0≤k≤n

Q: Let m, n >0 be integers. Prove that if m +n 2 29 then m > 15 Question 11 or n> 15 by contradiction.

A: To prove m+n≥29 then m≥15 or n≥15 where m,n≥0 be integers.

Q: Prove by induction E 3₁ = (3^+¹ -1) 2 i=o

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Q: 1. If n is an integer, then 2n +1 is odd.

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Q: 9.5 The celebrated golden ratio o is the positive real number number satisfying 1/o = $ – 1. Show…

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Q: Question # 03(c) Prove that neither 2 nor 17 ase Prime elements in Zli].

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Q: Question 6.7. Prove that x'(Km) = n if n is odd and x'(Km) = n – 1 if n is even without using…

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Q: 1. Give the direct proof of the following. a. If n is an odd integer, then n2 is odd.

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Q: Suppose n E Z. Prove that if n is even, then n2 + 5n + 4 is even.

A: Note: We know that the square of an even number is always an even number, since  Let x is an even…

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

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Q: Assume that S = Σ(-1)"-¹. 999 9 99 8 O 1000 10 1 (n + 1)5 n=1 n=1 Find the smallest integer N such…

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Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: QUESTION 9 Use Mathematical induction to prove that 2 divides n2+5n, n21.

A: Question 9: Let n=1 is true for Let the result be true for n=k,  We need to prove the result is…

Q: Prove the following using mathematical induction. 2. Šili+1)="(n+1)(n+2) for all neN. 3 i=1 3. For…

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Q: 4. With the fact of 1² +2²+...+ n² = n(n+1)(2n + 1)/6, prove that n n n(n −1) ΣΣ(i-j)² 6 i=1 j=1…

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Q: 9. Prove the formula n(n+1) Si=1+2+...+n= 2 i=1

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Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: Prove NIn, +oo[= 0 ! n=1

A: To prove that ∩n=1∞[n,∞)=∅. Suppose that x∈∩n=1∞[n,∞). Then, x∈[1,∞), x∈[2,∞),.... That is, x≥n,…

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: use mathematical induction to prove, for n > 1: 4n3 E(2k – 1)? = | 3 k=1

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Q: in is where n is an integer. e(1+4n)i where n is an integer. O equal to the above equal to the above…

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Q: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: 7. Prove by induction that E,(-1)² = (-1)"n(n + 1)/2 for all n > 1.

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Q: Prove the following using induction. 2k? = 1? + 22 + 3? + ... + n² = п(п + 1)(2n + 1) 6 k=1

A: We have to prove the following using induction -  ∑k=1nk2 = 12+22+32+........+n2 = n(n+1)(2n+1)6

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

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Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: 13. Prove E-1(4i – 3) = n(2n – 1) for natural numbers n > 1 %3D

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Q: Prove by the principle of mathematical induction that 3n? + 3n + 1 < 2" for n28 Ensure that you…

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Q: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

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Q: 1. Prove: If n is odd then 5n + 4 is odd.

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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: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

Q: 2. Prove the statement 12 + 22 +32 + 42 +..+ n : n(n+1)(2n+1) using PMI.

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Q: 6. Consider a = 7+ 2i and B = 3 – 4i in Z[i]. Find o and p in Z[i] such that < N(B). = Bo +p with…

A: Answer is in the next part. Note that as we are not aware of the construction you are talking about,…

Q: Let n € Z. Prove that if 3 /n, then 3 | (2n² + 1).

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Q: Prove the following statements by mathematical induction: - (a) Vn € N, - Σ (21 (2i − 1)² = n(2n +…

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Q: (e) The proof of Theorem 1 is thus currently incomplete, as it does not explicitly use the…

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Q: Prove that 2" > 2n Ior all n E N, n 2 3.

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Q: tn be a positive integer and M = [2 01 %3D en M" 2n equal to the above None of the mentioned

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Q: 9. Let n E Z. Prove that 3k EZ such that n' = 8k or n' = 8k + 1. In other words, prove that by…

A: We will use the fact that any integer is either even or odd to prove the required result.

Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: Let n e Z. Prove that if n is odd, then 5n + 13 is even.

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Q: 1) Guess a formula for the sum 1-3+5-7+..+ (-1)*-(2n – 1) =(-1)-(2i – 1) i=1 and prove your fornmula…

A: The given sum is  1-3+5-7+...+-1n-12n-1=∑i=1n-1i-12i-1

Q: Choose the answer that best completes the proof that inf(A)=0 where A = for all nEN n? Firstly, we…

A: Given the sequence A=1n2:n∈ℕ.  We need to prove that infA=0. The elements in the  sequence is of the…

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: 1. For In 5n2 + 9n3 + 1, ɑn =n², prove or disprove: In O(an) (n → ∞)?

A: Given:  xn =5n2 + 9n3 + 1, αn=n2. we have to prove or disprove : xn =O(αn ) ( n→∞).

Q: 3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides…

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