Prove that if m | k then 2" – 1| 2k – 1

Q: {p € Z :x |p}U{p €Z:y|p} C {p E Z : n | p}.

A: Number theory

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: Let n E N. Prove by mathematical induction that 3* = 4". k k=0

A: We have to prove given expression using mathematical induction.

Q: Let A = {n ∈ Z; n is odd} and B = {n ∈ Z; 3n 2 + 7 is even}. Prove that A = B.

A: We have to prove A is contain in B and vise versa

Q: 2. Prove that for any natural n the following holds 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n + 1

A: Note: according to our guidelines we can answer only second question  2)

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

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove that the statements here are true for every positive integer n. 1/1 . 2 . 3 +1/2·3·4…

A: Given statement is, 11·2·3+12·3·4+13·4·5+.......+1n(n+1)(n+2)=n(n+3)4(n+1)(n+2)

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

A:

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

A:

Q: Prove that the statements here are true for every positive integer n.   1 · 2 + 2 · 3 + 3 · 4 + · ·…

A: To prove: For every positive integer n, 1.2+2.3+3.4+.......+n(n+1)=n(n+1)(n+2)3

Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

Q: Prove that: n(n+ 1) r = 2 i=1

A: ∑i=1nr=1+2+3+....................+nhere it is a arithmatic progression sum of n terms of arithmatic…

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

A:

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

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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: Prove that for every positive integer n, 1 1+ V2 1 +...+ V3 1 > 2(Vn+1– 1) Vn

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Q: Prove that for all positive integers k < n, the equality n n n =2n-k k k i=k holds.

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Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: Let p be prime. Prove that if H ≤ G, |G| = 2p and H is not normal, then |H| = 2.

A:

Q: Let r(n) be the number of n-permutations whose square is the identity permutation. Prove that if n >…

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Q: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

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Q: Show that for any r, n > 1, п! Σ = r". nį!n2!...n,! nį+n2+…+n,=n, n¡ 20 11 (

A: Given information: The sample size is n. The number of categories is r.

Q: Prove that (n+1)C n is equal to n+1

A:

Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

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Q: Prove that, for all integers n > 1, 1+ 3+ 5+7+ ...+ 2n – 1 = n².

A:

Q: Prove by induction that for every integer n 2 1, (xe2*) = 2"-1(2x + n)e", ne Z

A: We prove this by Mathematical induction

Q: Prove that for any natural n the following holds 1 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n +1

A: We have to prove that for any natural n the following holds…

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: Prove that (n!+1, (n+1)!+1)=1

A:

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

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

A:

Q: Prove that if k <n and (a^k - b^k) | (a^n - b^n), then k |n

A: We will prove it using contradiction

Q: Prove that ℤ/n has exactly n elements.

A:

Q: Assume that n ∈ N. Prove that if 3|(2n + 1) then 9|(10n^ 2 + n − 2).

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n V3n

A: For all positive integers n, we need to prove that, 12·34···2n-12n&lt;13n

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

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Q: 18. a. Prove that for every integer a, if a' is even then a is even. b. Prove that V2 is irrational.

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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: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…

Q: 2. Let n be an integer. If x and y are integers such that n x and n | y, prove that {p € Z: x|p}U{p…

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Q: 3. Prove that, Σk() = n2n-¹. k=1

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Q: 8

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Q: 1. Suppose a e Z is an integer. Prove that a is odd if and only if a³ is odd.

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Prove that if m is a positive integer and x is a real number, then [m2] = [=] + z++ + 2 + |x + т — 1…

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Quadratic Equation

When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.

Demand and Supply Function

The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.