For every prime p, show that Х-1 — 1 %3 (х — 1)(х — 2) : .. [x — (р — 1)] in Zp[x].

Q: 7. Suppose I've defined a function f : N → N and I want to show that f(n) < 10n for all n2 5. To do…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: Find all n ∈ Z for which 3x^2 + 2nx + 12 is irreducible over Q.

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Q: n! diverges because an > 0. for all n > 1. True False

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Q: Construct a recurrence relation for T(n), with T(1) = 1, such that n² log2(n) 1.

A: Given: Tn=3n2log2x+1-n2log2n

Q: Verify by o Mathematical Induction dt todt E (2-1) = n²(2n²-1) for alt nz!

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Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

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Q: 2- Prove the following recurrence relation: Jn(x) = (-1)" J-n(x) By using the generating function.

A: As per our guidelines we are allowed to answer only one question. Kindly repost the other as…

Q: For any k e Z, give a formula for the coefficient of x* in the expansion of (2-!)" 100 x²

A: Given expansion: x2-1x100 The given problem can be compared to the general form x+an. Here, n=100…

Q: Let p be a prime. Show that there is exactly two elements a belongs to Zp s.t a^2 = 1.

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Q: Find f (n) when n = 4k, where f satisfies the recurrencerelation f (n) = 5f (n∕4) + 6n, with f (1) =…

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Q: diverges because an > 0. for all n 2 1. 2n + 1 True False

A: Given: The statement "an=n2n+1diverges because an&gt;0 for all n≥1." To determine: Whether the given…

Q: Let T(N) = 1. T(floor (N/3)) + 1 and 7(1) = T(2) Prove by induction that T(N) ≤ log N + 1 for all N…

A: Given: TN=TfloorN3+1 T1=T2=1   To Prove: TN≤log3N+1 for all N≥1, using induction.

Q: (b) Let Ho =1, H1 =3, H2 =9, and define Hk = Hk-1 + 3Hk-2 + 3Hk-3 for k23. Show by induction that Hk…

A: We will prove the result by induction  H0=1≤30 H1=3≤31 H2=9≤32 The result is true for k=0,1,2 Assume…

Q: Find an expression for sup in terms of inf E, sup E, inf F, sup F and prove it. 2F

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Q: -), prove that for all n E N: (a) n-1 E(:)-E- n k k=1 k=0 (b) ¿(:)-- = 2" k k=0 (c) El-1)* E-1» (:)…

A: According to Bartleby policy we solve up to 3-subparts. Please repost #d and mention it.   b)…

Q: Let n be a positive integer. The coefficient of x°y° in the expansion of (1+ xy+ y²)' equals

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Q: 1. Let, a1 = 7 and for n 2, an An-1 express an in terms ofn. %3D ". **

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Q: Find the elements (a, b) of G of order 2. O Show that G has no element of order 3.

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Q: Determine the z-transform of the following: n 20 x(n) = n<0 O · </²/<2 5 3 1-²-(1-22-1) 5 3 1 -z 3…

A: Z transform of the function

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

Q: For m an n non-negative integers with 0 < m <n, give a combinatorial proof for the identity EOC) m…

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Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: d Use induction to prove that (x") = n· x"-! for all n e N. dx

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Q: -1" n! 1. Prove that S xm (In x)mdx (m+1)n+1

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Q: Determine the z-transform of the following: [信) n20 x(n)= 18 nco 5 <2 (1-2z-1) 5 --Z -1 공시디 2…

A: Z transform of the function

Q: The first term in the Maclaurin's expansion of f(x)= In(1 – 2x) is O-2 o not defined 0.

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Q: Prove T(n)= 2T(n-1)+n can be in 0(n2")using represented telescoping method

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Q: Consider a population that grows according to the recursive rule Pn Pn-1+ 90, with initial…

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Q: Let x1 V3, xn+1 = V3*" ,n > 1. Use induction to show that x, < 3 for all n E N.

A: Given x1=33,  xn+1=3xn3. Show that xn≤3 by using induction for all n∈ℕ.

Q: Let c be a number such that |c| 0. Then use the binomial theorem to show that 1 |c"| < 1+ nd 1 nd…

A: According to the given information, Let c be a number such that:

Q: Show that t (T (n)) = 2 if and only if n = p9-1 for some primes p, q. %3D

A: T(n) is the divisor function which counts the number of divisors of a number.

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

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Q: Let the characteristic of F be n =/0 . Then n is prime.

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Q: m Show that E (")C")= ("*"). Using Combinatorial Proof. \m+p. k=0

A: We will use equating coefficent method

Q: Give a combinatorial proof: Forn and k satisfying 1 <k < n, n nk = Ek · (j – 1)k-1. j=k

A: The complete solutions are hoben below

Q: p be an irreducible reprsentation of G

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Q: Is the stries cm vergent or divergent? (Ghow how you kenowi) 2 3n n'+ In + 13

A: a The given series is ∑n=1∞23n.We know that the series ∑n=1∞1n is divergent.And…

Q: If X ∼ P oisson(λ), Prove that for any positive integer k, E[X(X − 1)(X − 2)· · ·(X − k)] = λ^ k+1

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Q: {3* + (-3)k + 3k}

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Q: n! diverges because an > 0. for alln2 1. True False

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Q: 2. Calculate Fn, when n=0, using the Binet form

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Q: If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

A: Given: G=p.q where p &amp; q are primes To prove: ZG=1 or pq

Q: In a Wilcoxon Mann Whitney test n1 is 9 and nprime is 7. What are the correct values for K1 and K2 ?…

A: In a Wilcoxon Mann Whitney test n1 is 9 and nprime is 7.  We need to find the correct values for K1…

Q: 1. Find the first-order derives:

A: Given,      4x2+715x4+534x3-37x6+15x-30

Q: 4. Prove that the L{2t²} = 4 S3

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Q: The first term in the Maclaurin's expansion of f(x) = In(1 – 2x) is

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Q: Let ain be integers with 2<n ond ecd (ain) - d. Show that if ok #I lmodn) for all k E { pln)-} then…

A: We know that Uℤn is the set of all units of ℤn. First let k=1, then ak=a1=a. Given that gcda,n=1.…