Let i be a positive integer and A, = [-i+1,1]. Then U, A O R O 1-0,1] None of these -1,1]

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

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: n -r Prove that („",) : r+1

A: Combinations: Combination is the selection of n items taken r at a time is

Q: Prove that Eo 2i = 2n+1 – 1 - i3D0

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Q: Prove the identities. 9n Afn-fn A9. In gnti %3D gn

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Q: 4. Prove by mathematical induction that for every positive integer n, II (G- 1 %3D 3i-1 31 3i+1…

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

Q: Theorem 23. If u, v E Q then we can find integers a, b, d with d > 0 such that u = a and v = d'…

A: By Division Algorithm we know that,               Let a, b be two integers with b>0. Then there…

Q: Q1. Prove that for any natural number n, E, 2i = 2n+1 – 1. i%3D0

A: I have used mathematical induction to prove the result

Q: Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

A:

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: Let 0 < a < 1 be a real number. Define a₁ = 1+a, Prove that Vn € N, 1 ≤n, an+1 = 1<an< 1 - an + a, -…

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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: (b) Let A¡ = [-i, 1] for every positive integer i. Find U A¡ and N1 Aj.

A: Given that Ai=-i,1 for every positive integer i. That is, A1=-1,1, A2=-2,1, A3=-3,1 and so on. To…

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 A=I-N+N²-N³ for A & N definec as follows: N=A-I (identity), where |1 0 0 an & I= 0 1 0. 0…

A: given matrix  A=1a12a1301a23001 and I=100010001 prove  that A-1=I-N+N2-N3 where N=A-I

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

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Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Let n be a positive integer. Let H = {0, ±n, ±2n, ±3n, . . .}. Findall left cosets of H in Z. How…

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Q: Determine whether 3 and 5 are irreducible or not in Z[i]. 3 and 5 are reducible. O 3 and 5 are…

A: We have to check

Q: Let {Xi} be i.i.d.∼ Exp(θ) and ai > 0. Denote by Z = min{a1X1, . . . , anXn}. Suppose that sum 1/ai…

A:

Q: Let i be a positive integer and A; =[-i+1,1]. Then U A;= %3D O 1-0,1[ O [-1,1] O R O 1-0,1] O None…

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Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

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Q: Let and 001 1 0 0 2 1 A = 0 If the entries of A and v belong to Z3, what is the third entry of Av? 2…

A: Recall matrix multiplication and modulo arithmetic.

Q: Prove that x' +4=(x+1+i)(x+1-i)(x-1+i)(x-1-i).

A:

Q: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

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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: List the elements of the power set of N and enumerate all o-fields containing {0}. 1. 0 = {0} %3D 2.…

A: Sigma field is closed under countable union and complement and the empty set always belong to a…

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

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: Show that if n = p1p2⋯pk, where p1, p2,…, pk are distinct primes that satisfy pj − 1 | n − 1 for j =…

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

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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 by induction Ein (-i)i ;2= (-1)" n(n+1) i=o 2

A:

Q: (a) If an+1 = an/(an+1) prove that an = ao/(nao+1) for all n ≥ 0. OL

A: Note that this ques are all separate questions and not related to each other. Just the process is…

Q: part ii and iii, thank you

A: According to pascal law,

Q: Let 0 1

A: Introduction: A strong and elegant method for proving specific kinds of mathematical claims is…

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: I need the answer as soon as possible

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Q: Let i be a positive integer and A; = [1,i+1[. Then U A OR O [-1,1] O [1, [ O 11, [ None of these

A: [1,inf[ is correct answer

Q: Let yII + yI - 42y = 0 Find all values of k such that y = ekx

A:

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: 1) Prove that 0|a - a=0

A:

Q: Prove that 1 n i(i+ 1) n+1 i=1 whenever n is a positive integer.

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Q: 5. Evaluate the integral JI. dA, R = [-2, 4] × [1, 3]. R

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Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: Show that Vn 2 0 E-o(2i + 3)3 = (n + 1)3"* %3D

A:

Q: 12. Prove the following (A+ BB")-B= AB(I+ BT AB)-1 by express the B+BB' AB in two forms. O Type here…

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Q: 1. Let A = (ai)nxn be symmetric and positive definite. Show that a) a>0, i=1,2,...,n. b)…

A: The given matrix A=aijn×n is symmetric and positive definite matrix. To show: (a) aii&gt;0 ,…