10. Show that for all n E Z, n² – 3n +1 2.

Q: Consider a, = V4n² + 3n – 2n for n > 1. %3D -

A: Introduction: The sequence is a set of functions with the set of natural numbers as the domain of…

Q: 6. Prove that for x > -1 then Vn E N, (1 + x)" >1+ nx .

A:

Q: Prove if equal to MPn (W) dx= 2" (n!)" (2n+)! that m is an integer

A:

Q: п-1 k n-2 n-1 n-2 () x6n-10 = p %3D k - 4 X6n-9 = 4 n-1 n-1 n-1 k n-1 X6n-8 = h X'6n-7 = P k n-1 n-1…

A:

Q: Suppose that x̅ = 12, Sx = 2, and z is 1. what is X?

A:

Q: 2. Use the Root Test to determine absolutely or diverge. 4n+3\ n (a)E-1( Zn=1 Зп-5) (b) E (-In (e²…

A: As we have to use the root test , So first I have written the Root test then using them for both the…

Q: What can you say about the upper bound of E(x) for numbers of the form 3^(2n) + 1?

A:

Q: 1 О. 15 Show that %3D (D - Q) k e k

A:

Q: Q3: Prove that (9":n e Z} S {3":n Ee Z} but {9":n e Z} (3":n E Z}.

A: To prove:  {9n:n∈ℤ} ⊆{3n:n∈ℤ} but {9n:n∈ℤ} ≠{3n:n∈ℤ}

Q: Prove that L ={(1Z*)"10" | n e N}is not regular.

A: The given language: Then, the above language can be rewritten as, To prove that L is not regular.…

Q: 2. Find p and q such that |5x - 2| ≤4 implies Ps 23x-4 5 ≤q.

A: Given |5x-2|≤4

Q: Show that for any n E N, 7(15".

A:

Q: 8a, 2+ 225n for n> 2 with initial conditions ao 7, a1 83 %3D

A: Given: an=-6an-1-8an-2+225n for n≥2 with initial condition a0=7, a1=83. To find: The solution of the…

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

A:

Q: 3. For all æ E Z[i], there are N (x) elements in Z[i]z.

A: The set of Gaussian integers is a subset of the set of complex numbers containing the elements with…

Q: Show that for any 1 ≤k≤n. T (1) = n²(n²-1²)...(n² (k − 1)²) 1.3.5 (2k-1)

A:

Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

A:

Q: If E(0) = 0 then is said to be a(n)

A: if Eθ^=θ then θ^ is said to be unbiased estimate for θ.

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: Suppose we observe X1, X2, ..., X U (0, 1). Show that nX(1) 4 Exp(1). с.

A:

Q: 4 16 Define T: R? - R° by T(x) = Ax, where A = 1 3 -

A:

Q: Show that for any n E N, n(n + 1)(2n + 1) k=0

A: The solution is given as

Q: 5.7.4. Show that the two formulas (r = y) and (S(r) = Š(y)) in N.

A: Hi, since there are 2 questions given, the first one is answered.  Given - The formula  x=y To prove…

Q: Prove m3 – n° = (m – n) (m² + ²). - N m - n (m – n)(m² + mn + n° Enter your next step here %3D 2.

A:

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: Show that for all n E Z, n² 5.

A: Consider the given condition: This shows that whatever will be the values of n, it will always be…

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

Q: (b) f(z) = tan¯"(5z) - (In(2"))®

A: We have to find derivative

Q: (5) Show that T(n)o(n) > n for all n E N. 11

A:

Q: Find L {(2 t" – 3) (t +2)} O 2(2n)! n! + s2n+1 6 sn+1 2 (n²!) n! n² +1 sn+1 O 4 n! n! + s2n+1 8n+1 O…

A:

Q: Prove for r ≠ 1 that ∑_(k=0)^(n-1)(rk)=(1-rn)/(1-r). If ∑_(k=0)^(n-1)(rk), then compute s(1−r) =…

A:

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: () 2k = 3", Vn E N k k=0 k n+1 Σ 2 Vn E N, n > 1 3 k=2 į * () 主) n 3k = 4", Vn E N %3D k k=0 = 2"-1,…

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

Q: 15. prove that L. d= nd(n)/2

A:

Q: 2. Show that coo(N) co(N) § c(N) Ę e^(N).

A: To prove that c00N⊂c0N⊂cN⊂l∞N. By definition, c00=x∈lp|all but finitely many xj's are zeroc=x∈l∞|xj…

Q: n2 for all n E N. 4n Let r, :=

A:

Q: 2.4.10 Show that det(e4) = etr(A) for all A E Mn(C). D %3D

A: The given problem is to show the given identity. To show: det(eA)=etr(A) for all A belongs to Mn(C).

Q: Let A = {10k:k EZ*} and M = {5k:k e Z*}. Then None of these Ο Μς Α MC A O AC M O A=M

A: Solution:

Q: 20 + 7)* Evaluate the integral dæ + 7)* * by making the appropriate substitution: u = dx (2x + 7)4

A: Let's find integration of given function.

Q: 1. For an = 5n2 + 9n3 + 1, an = n2, prove or disprove: rn = O(an) (n → o)?

A: We will use the standard definition.

Q: es lls ew zeiflucifiib ons einiseen stigesber Show that for neN, (2n)! ' (r!)² ((n −r)!)² - 7=0 2n -…

A: We will use binomial expansion to prove given relation

Q: Suppose X~N(113,18), and Z~N(0,1). Then x = 94, is equivalent to z = ______.

A: Given that, Suppose X~N(113,18), and Z~N(0,1). Here, mean μ=113 and standard deviation σ=18

Q: Show that for any n E N, n(n + 1)(2n + 1) %3D k=0

A:

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: 26. If z = In(x² +y²), find x- ду Əx A) х с) у в) х-у D) 0

A: If z=ln(x2+y2), We need to find x∂z∂y-y∂z∂x.

Q: = 14 |1 + 0 (())| for all i with (b) n = 4 [1 + 0 ()“)- (a) qki (x1); + 0, 2k

A: a) Given that λ1>λ2≥…λn Now,  x1Ty0 is not equal to 0 Now, expanding it  yo=a1x1+…anxn so that…

Q: (3b) Let n E Z. Prove that if 3|(2n² + 1), then 3 ł n

A:

Q: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

A:

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

A: