1. and for n 2 2, The Fibonacci numbers are defined as follows: fo = 0, fı = 1, and for n > 2, fn = fn-1+ fn-2. Prove that for every positive integer n, f3 + fo + ..+ f3n =(f3n+2 – 1) – 1)

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: The notation x = y mod n is used to indicate the unique integer x in the range 0 :$ x < n such…

A: Given statement is that the notation x = y mod n is used to indicate the unique integer x in the…

Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

A:

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: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: Let n∈Z.  Prove that x is odd if and only if 5x + 4 is odd.  Identify the type of the above…

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Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

A:

Q: 1. In the proof of the claim "there is no integer n such that n = 2 mod 6 and n = 7 mod 9," the…

A: See answer below

Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

A:

Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

A:

Q: Find k such that, 2 det 4 li0 -8 -5] -1 -7 = 510 k 0 .

A: Given det2-8-54-1-710k0 =510 we have to find k

Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

A:

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

A:

Q: 4. Let bo = 1, b1 = 3, and for cach integer k> 2, we have bz = 4bg–1 - 4bg-2 Prove that for all…

A:

Q: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: 5. The Fibonacci numbers are defined as: fo = 0 , fi = 1 , and fn = fn-1 + fn-2 , for n > 2. Give a…

A:

Q: 4. Let bo = 1, b1 = 3, and for each integer k > 2, we have br = 4bk-1 – 4bg–2 %3D Prove that for all…

A:

Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

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

A: We will solve the problem

Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

A:

Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: Compute the infimum and supremum of the set {n+(1/n)" : n E N} if they exist.

A:

Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

Q: Is the following statement true or false? For every odd integer n, To answer this question, let n be…

A: Let, n be any odd integer. By definition of odd, n=2k+12, for some integer k So, by solution,…

Q: II. Prove the following using Direct Proof. 6. If x is an odd number and y is even number, then x +…

A:

Q: 4. Let bo = 1, b = 3, and for each integer k > 2, we have br = 4bx–1 – 4bk–2 Prove that for all…

A:

Q: Show that if M is a number less than 1, then the terms of {n/(n + 1)} eventually exceed M. That is,…

A:

Q: 2. For each of the following, use a counterexample to prove the statement is false. * (a) For each…

A:

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: 12. Let the "Fibonacci-2" numbers g, be defined as follows: 91 = 2, 92 = 2, and gn+2 =( In+1)( In)…

A:

Q: 4. Prove that f+ f+..+f = fnfn+1, where n is a positive integer and fn is the nth Fibonacci number.

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Q: Let the numbers c1, C2, ... be defined inductively as follows: c1 = 0 and for every n >1 the number…

A:

Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: Prove directly from the definitions that for all integers the Same parity Divide Bnt 19. and E m and…

A: If m an n are both even  let m= 2a and n= 2b   then  5m+7n=5(2a)+7(2b)=10a+14b=25a+7b therefore,…

Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

A:

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

A:

Q: Prove the following using the appropriate method.    Let a, b be integers. If ab is even, then at…

A:

Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

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Q: Prove that for any positive integer n, there is a k e Z>o and ao, a1, . .. , ak E {0,1} such that k…

A: Please check later steps for detailed answer

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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Q: 1. Find a number that provides a counterexample to show that the given statement is false: For all…

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Q: For any integer n2 1. prove that: (a) 1+2+3+..+ (n – 1) + n + (n – 1) + . .. +3+2+ 1 = n². (b) +...+…

A: Note: According to Bartleby guidelines; for more than three questions asked, only the first three…

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.