2. Prove by principle of Mathematical Induchion 6.7h - 2.3n is divisible by 4 for all nE 2#. 1

Q: Prove the following fact: A" = 3"-1A. 1 1 1 1 1 1 1 1 1 A =

A: Given, A =111111111  We have to  prove An = 3n-1A

Q: &2/ suppost se =20, 1} and c={208{ fie ld =20,18 and e= Enumerate the class of all 6- Containing C…

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Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

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Q: (2) Prove that 21+ + for all 21, 22 € C.

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Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

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Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: 6. Prove that if gcd(a , b) = 1, then gcd(a + b, ab) = 1.

A: We need to prove that if gcda,b=1 then gcda+b,ab=1 Let us on the contrary assume that  gcda+b,ab=c…

Q: is an element in X that is not in Y. O No element in Y is in X. O There is an element in Y that is…

A: We will determine the meaning of subset and solve the problem.

Q: 5. Prove that S(n +1, k+1) = £ $(m, k) (").

A: To prove- Sn+1,k+1=∑m=knS(m,k)nm The falling factorial is defined as the polynomial,…

Q: 3. prove that: det (AB) = det (A) det (B) If a - (;). » - ( :) 2 -1 7 A = 3 2 4

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Q: Express in the form & j(n/2-j) (2) In(3

A: As per the bartleby guidelines for more than one part only one has to be solved. Please upload the…

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

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Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: 4 Show that a + B = B +a for all a, BE C.

A: We can write alpha and Beta in general form of element of complex number.

Q: Prove that in Q(/-5) we have (3,1+ 2v-5)(3, 1 – 2-5) = (3).

A: To show the equality, first we take an element of the product and show that it is in (3) and then we…

Q: Prove that in Q(V-5) we have (3,1+ 2/-5)(3, 1 – 2v-5) = (3).

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Q: 3- Compute (3-5*)(J5li +3). any leave your answer in sfandard fam

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Q: 1. Prove that In(1 + 1) <I for r 20.

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Q: 2.1. Prove that B Is erse of A -17 11] -7,B: 0 3 2 [1 1 2 2 4 -3 [3 6 -5] 2 A = -1 11

A: To prove that B is the inverse of A.

Q: Write down the supremum of the set {7+ 7/n2 − 5/m | n,m ∈ N}. prove ur answer from the definition.

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Q: 2) Prove that vnEN 12n+1 is in lowest term, i.e., prove that 30n+1 gcd(12n + 1,30n + 1) = 1 3) Vns A

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

Q: (" * ?) Prove: 1n + 2(n – 1) + 3(n – 2) + …+ (n – 1)(2) + n(1) = using lattice paths.

A: As per the question we have to prove the following summation identity : 1n + 2(n-1) + 3(n-2) + ... +…

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

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Q: Prove that if n 2 3, the only element o of S, satisfying oy =ya for all y E S, is the identity…

A: Let n≥3. Let σ∈Sn such that σγ=γσ for all γ∈Sn. Suppose σ≠i, the identity Then there exists…

Q: In AABC, m2C = 90° A, Z1 and 22 are complementary Given: Prove: 2

A: Sum of measurements of angles of a triangle is 180 degree.

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: 2-Prove that: B (m, n)-ß (a,m) 3- Simplify to the simplest: (2+0(1-0) (1+0)(3-2)

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Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: Let n∈Z. Prove that n3−7n+18 is divisible by 6 for every interger n. Do not use induction.

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Q: discrete

A: In method of contradiction we start by assuming that given statement is false. Then prove that it is…

Q: 4. Prove or provide a counter example for the following: (AU B)nC = (AnC)U(BnC)

A: We need to prove that(A∪B)∩C=(A∩C)∪(B∩C)If we have to prove that(AUB)∩ C C C…

Q: 7. Prove that 2. O = (-1). %3D

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Q: is a1 = 2, a2 =3, and an  = 5an–1 – 4an–2 for n > 2 expressed in explicit formula? true or false?

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Q: II. Prove the following. 3. 7|23n – 1, Vn EN -

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Q: Prove that: "C. = 2" {1.3.5...(2n - 1)} n!

A: The left-hand side of the expression is: lhs=Cn2n=(2n)!n!(2n-n)!                               Use…

Q: (3) prove That () 3

A: We have to use z = x + iy.

Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…

Q: 3. prove that: det (AB) = det (A) det (B) If 2 -1 7 2 B = A 3 2 -3 4

A: In the question it is asked to prove the property det(AB)=det(A).det(B). Where A=2-132 and B=72-34

Q: 3.1 State the Weierstrass M test theorem.

A: Note: As per our guidelines, we will solve only first question. If you want any specific question to…

Q: 2. Prove that Qn [2, 3] ~ N.

A: Two sets A and B have the same cardinality if there exists a bijection (onto , one-to-one…

Q: @ 2 show Hhat fox) = e*-x3+2 2+2 is a %3D Permutation on R.

A: In the give question we have to show that the function f(x)=e-x-x3+2 is a permutation on ℝ.…

Q: 4. Prove that it cannot be the case that x EA\B and x EB\C. (Hint: prove via contradiction).

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Q: Prove that 4 + 8 + 12 + .... + 4n = 2n (n + 1 )

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Q: 12.42. Let a, be Z, where not both a and b are 0. Prove that if d = gcd(a, b), a = ajd and b = b1d,…

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Q: 2. Expand the following: (i) apxi (ii) 7 (√ga') (iii) A Bi

A: To expand: (i.) aijxj (ii.) ∂∂xigai (iii.) AikBi

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

A: We have to find solutions

Q: Prove that ∀n ∈ Z, gcd(7n + 1, 15n + 2) = 1.

A: