Prove that 3 | 4" – 1 where n E N.

Q: Use a proof by contradiction to prove the following: П For all 0 € R, if 0 1 2

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Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: Prove that Σan σ(d) φ (n/d) nτ (n) and ΣIn τ(α)φ(n/d) σ(n). ||

A: As per the question we are given two sum identities including the Dirichlet convolution :…

Q: Prove that the sum of (") where i is from 0 to n is 2".

A: According to the binomial theorem, 1+xn=n0x0+n1x1+n2x2+⋯+nnxn……1

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

A: Solution:     Let 3 does not divide n      => n = 3k ± 1 for some integer k.       => 2n2 + 1…

Q: nl diverges because an > 0, for all n 2 1. ) True False

A: Given statement is

Q: (b) Show that o = 2[Rx(0) - Rxx(0)] XX

A:

Q: Q3: Prove that (9":n e Z) (3":ne Z) but (9":n E Z) (3":n E Z).

A: We have to prove that 9n:n∈ℤ⊆3n:n∈ℤ but 9n:n∈ℤ≠3n:n∈ℤ

Q: Q5 Show that EL >². n n =1

A: Consider that x is a random variable. From the formula of population variance;

Q: 1. • If r + 1| < o, prove that |r³ + 1| < 0.331.

A: Here we use the fact  | x |&lt; a implies  -a&lt; x &lt; a .

Q: - such that n >r > 1, (n – 1)! r!(n – 1 – r) n! r!(n – r)! n - r -

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Q: d x" = nx"-1 de 4. Prove that NɛQ

A: ddxxn=nxn-1 ;   n∈Q

Q: What is the general rule to evaluate I N,where N is any positive

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Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

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: that Edn H(n/d)7(d) = 1 and Edn H(n/d)o(d) = n for all n, where r(n) = d(n) and σ(n) = σι (n) Σm d.…

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Q: Prove that n2+ 4n + 6 E O(n²)

A: Solution: The objective is to prove that the given statement

Q: nº +n² + 1 4. Prove that = O(n³)

A: To prove: n4+n2+1n=Θn3

Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

Q: (a) Prove that for all n 2 0, Eo fi = fn+2-1 where fi-1 is the th Fibonacci number.

A: Fibonacci sequence

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

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Q: Suppose X~N (μ,σ') then X – H - - N (0, 1) o2 O True O False

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Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

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Q: Prove indirectly: "V n E Z, 4 ł (n² – 2)."|

A: Ans:Let us assume that 4/n2-2;∀n∈zthen we have there exist a positive integer k such…

Q: Verify by mathematical induction Ž(2i-1 1. 1 - n° (2n? – 1) for all n 2 1. n(n+1) 2. Prove: V n € Z,

A: As per our guidelines, we are allowed to answer only one question. Since you have asked multiple…

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

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Q: O N (AT) = R(A)-

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Q: Suppose x, n ∈ N and n not equal to  0. Then, show that xn is positive if and only if x is positive

A: Given that x, n∈N and n≠0. According to question, N=0,1,2,3,4,5........ First suppose xn is positive…

Q: 17. prove that Edln H(n/d)7 and σ(n ) σ(n) Σ %3D

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

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Q: Show that for any n EN where n > 1, II (1 = n + 1. k k=1

A: We need to prove that for n≥1, ∏k=1n1+1k=n+1 We will prove by the Principle of mathematical…

Q: If nis a positive integer 1 then prove (D - a) n!

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Q: 2 (x) 2. prove that s

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Q: n(n – 1) is O(n²). 2 Show that

A: f(n) is O(g(n)) if there exist positive constants c and k such that f(n)≤c g(n) whenever n&gt;k.…

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: Prove that tanodo-n//2.

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Q: 5. For non-negative integer n and r, where 1<rsn, show that (n-r+1)"C,1=n(""C,). r-1

A: Given n and r non negative integers with  1≤ r≤ n. We need to prove (n-r+1)nCr-1=n(n-1Cr-1)

Q: Suppose m=nd where n is odd and 1<n<m. Prove that 2m+1 is composite.

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Q: Prove that 2" > 2n Ior all n E N, n 2 3.

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Q: 8. Prove 3n e R|n? – n < 0

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Q: Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.

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Q: Find integers s and t so that s· 201 + t · 302 = 1 . S %3D

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Q: 7. f(t)=3e" cosh (In 21) -2t in two ways.

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Q: Use Euler-Maclaurin's formula to prove that E n² = "(n+ 1)(n +2) 6. %3D 1

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Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

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Q: (-1) is where n is an integer. equal to the above e(1+2n)i where n is an integer. equal to the abov…

A: We know from eular formula e^iθ = cos θ + isin θ

Q: Q1. Prove that KT = = -.

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Q: Ql/prove that : 11=00 H(elw) = Σ 71=-8 h(n)e-jwn

A: "Since you have posted multiple questions, we will solve only the first question for you." To Prove:…

Q: Show that yn Z, if n³+5 is odd then n is even

A: