1 Let consider two natural numbers m, n. Prove that a) m +n E N b) тxпEN 2 Prove that n + 4n" + 6 EN if n, m E N. 3 Find a condition on n EN such that 2n? -7n -4 EN.

Q: 5) Prove that : tanz 올n(를) i-z

A: tan-1z=i ln[(i+z)/(i-z)]/2.

Q: 2. Let N = {A, | ne N}, where A, = {n,n+ 1, n +2, ...}. (a) Find UN and prove your conjecture. (b)…

A: we have given Ω = An n∈N where An =n, n+1, n+2,...... we have to find (a) find ∪Ω and prove your…

Q: 1), n = 1, 2, . .

A: In this question, the concept of mathematical induction is applied. Mathematical Induction Inductive…

Q: 2. Prove that, for all natural numbers n > 1, 2+ 2° + 2* + 2ª + · + 2" = 2"+1 – 2. %3D ...

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Q: (5n)! 40"-n! is natural number for every natural number n. Prove that

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Q: Let n>,2 and k be any boretine Proce that integers (n-1)* | n, 2

A: Given, n≥2 and k is any integer. n≡1modn-1Therefore, ni≡1modn-1 ∀i∈ℕ…

Q: Prove: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

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Q: 2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction…

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Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

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Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

Q: Use mathematical induction to show that E"-, ar ark-1 a(1-r") for all integers n 2 1-r 1. (Assume r…

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x &gt; – 1.

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Q: Show that the set of a+bi : a€Z, b€Z, i=root of (-1) Gaussian integers has the same strength as the…

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Q: 2"-1 for positive integers n.

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Q: 4. Prove by mathematical induction that 20+2¹ +2²+...+ 2 = 2n+1 -1.

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Q: 8. Prove using induction that E-oG) = 2 1 k=0 - 2n

A: We use the principle of mathematical induction to prove the given statement.

Q: Show

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Q: A sequence {un} is such that un+3 = 6un+2 - u1 = 2, u2 = 6. Prove by mathematical induction that Un…

A: See the attachment

Q: Abstract Algebra

A: To find non-negative representations under the given conditions

Q: Q prove that for ony natural number n,2+4 +2n=1/n+1) %3D

A: The solution is given below.

Q: Prove that n and m are natural numbers such that n > m, then n-m is also a natural number. Prove by…

A: Suppose that n and m are natural numbers such that n &gt; m. Then we have n - m &gt;0…

Q: 11**) (Solution to be discussed in tutorial only) For a polygon to be conver means that all of its…

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Q: Prove there are no natural numbers n and p with n 2 2 and p² + 1 = 2".

A: prove by contradiction:- to prove it by contradiction, you should start by assuming its negation,…

Q: (ii) Prove, by induction, that F, × F, × F, × ...× F-1 = F, – 2. %3D (iii) Use part (ii) to prove…

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Q: Prove by induction that E-,(i × 2' +1) = 4(n × 2" – 2" +1). 'i = 1

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Q: #3. Let n be an integer with nz1 and Let b, bz . bn be ... positive real numbers, not all egual,…

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Q: Prove that ∀n ∈Z, gcd(7n + 1, 15n + 2) = 1 *use only the definition of divisibility and gcd please!

A: we want to prove that   gcd(7n+1 , 15n+2)=1

Q: Suppose (an) is Cauchy and that for all m, k ≥ 100, |am − ak| 0. Does it follow that the interval…

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Q: #3. Let n be an mteg er with nz1 and Let b., bzba be ... positive real nmmbers, not all egual, such…

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Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

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Q: Suppose (an ) is Cauchy and that for all m, k ≥ 100, |am − ak | 0. Does it follow that the interval…

A: Consider an is Cauchy and that for all m, k&gt;=100, |am -ak|&lt;1/4.

Q: Prove Osasb→0sVas15 Hint: Use proof by contradiction

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Q: 2. The binary operation * defined on Z by x * y = 1 + x + y on the set of real numbers. Show that *…

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Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

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Q: Prove there are no natural numbers n and p with n > 2 and p² + 1 = 2".

A: ANSWER:

Q: 2. Prove that võ is iational number. 3. Ifa,beR. prove that ab

A: ANSWER 2:

Q: Let min EN 7 atZ be such theut Ca,m)=1, Ca,n)=| ¥ min)=\, Let K = Om (a) I j= On(a). 31K that ak=I…

A: Given that m,n∈ℕ and a∈ℤ with a,m=1, a,n=1 and m,n=1.  Suppose  k=Omaj=Ona With j|k To prove that…

Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: Use induction to prove for all n eN that 2* = 2"+1 – 1. %3D k=0

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Q: Prove by induction that ∑_(i=1)^n▒〖〖2i〗^2+5i+2〗=(4n^3+21n^2+29n)/6 for positive

A: We will prove by induction  ∑ i=1n2i2+5i+2=4n3+21n2+29n6 For n=1  ∑i=112i2+5i+2=4+21+296 2+5+2=546…

Q: 4) Prove by induction on n that, for all positive integers n: 6(5n6" – 6" + 1 i6' = %3D 25 i=

A: # Prove  ∑1=1n i 6 i=65n.6n-6n+125      Let Pn be the proposition given above Base case:-…

Q: A sequence of numbers co, ¤1, X2, .. . is such that xo = 4 and, 6. %3D for each n > 1, 1. In 3n-1+2.…

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Q: [Hint: It suffices to show that the numbe 11. Verify that 0, 1, 2, 2², 2³, . , 2° form a complete…

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Q: Compute the supremum and infimum (if they exist) of the set A = {em-nl: m,n EN}. If they don't exist…

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Q: 3. a) Compute Ek. b) Prove that there are infinitely many prime numbers. c) Assuming modulus 127,…

A: Given: The following conditions. To find: a) Compute ∑k=0nk3 b) Prove that there are infinitely many…

Q: Prove by induction: 1- 3h 27 3* 2

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Q: 4. Prove by mathematical induction that = 2+1 1. 20+2¹+2²+...+2"

A: We have to find solutions

Q: #2

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.