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

Q: n = n and n = 1. n Show that in

A: A factorial is the product of consecutive integers starting from 1. It is known that nr represents…

Q: Let n E N*. a. Let a > 0. Then 1 as n 0o. b. - 0 as n → ∞ n16/5+2 c. (-1)"+1"" → 0 as n → 0 d.…

A: We know that limx→∞1x=0. It implies that if x→∞, then 1x→0. We need to calculate the limit for every…

Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

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Q: If xn → 0, then for every e > 0 there exists N e N such that n > N implies xn 0 there exists N € N…

A: Hey, since there are multiple sub parts posted, we will answer first three sub parts. If you want…

Q: For any n e N, let Bn = (n, 0). {Bn}=1 is descending. Show that N Bn = 4. n=1

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Q: Show in detail that E [(X-µ)^n] = 0 for ∀n , where X~N (µ, σ²) and n≥3.

A: It is an important part of statistics. It is widely used.

Q: (b) Show that k-1 (k+1) n n+1 for every n ≥ 1.

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Q: Show that inf n e N} = 0. n1/3

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Q: If (ar + br) 4n and n+1 k-1 E(ak - br) 1 then ar = | n+2 k-1 k-1 3 5. None 2. 72n/2

A: use elimination method Find limit for n tending to infinity.

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: Question 3. In parts (a) - (c), prove that for all n ɛ N: (a) n-1 £(:)-I- n k k=1 k=0 (b) £(:)-- n =…

A: Here we will use binomial series expansion  a+xn=n0an+n1an-1x+n2an-2x2+......+nn-1axn-1+nnxn…

Q: 3. Let A={n €Z, 2Z). Prove or disprove: For every n E A we have n² = 4.

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Q: 3 a. Suppose h(n)= 15n° + n and f(n) = 14n°. Is f(n)EN(h(n))? Use the definition of Omega notation.

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

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…

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Q: 3. Given n > 1, let S = E-1: Show that S, is not k=1 _k° an integer.

A: We are given that   Sn = ∑k=1n1/k   and n>1              ..(i) We have to show that Sn  is not an…

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: Prove from the definition: n3 + 65n2 + √n = Θ(n3)

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Q: (Зп+5 Find infimum of A |n E N} n+9

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Q: for all integers n >1 E, i = n(n+1} %3D 2

A: To prove the sum of n natural numbers n≥1 is  ∑i=1ni=n(n+1)2

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

A:

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

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

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Q: Fix k, n E N with n > 2 and k <n – 2. Prove that: | )-(".) · ) (:) ) (^) = (":") 2 + k 2 n – 2 n n +…

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

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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: For all n E N, let An = {0, 2n, 4n, 6n, . . {x € Z|(3n E NU{0})(x = 2n)}. (a) Find (with proof) [U…

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Q: Show that E [(X-µ)^n]=0 for ∀n, with X~N(µ, σ²) and n≥3 being odd numbers.

A: We need to show; E[(X-μ)n]=0 for ∀n and n≥3 being odd numberswhere;X~Normal(μ,σ2) That is, we need…

Q: Prove m+n-1 Σ k k k>0 k

A: We know that  nCr=n!r!(n-r)! When k= 0, the LHS becomes 0 When k=1, the LHS becomes…

Q: For any n large enough, prove E" ) – = 0(1). k=0

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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: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

A:

Q: at k ( k) = n(k-₁) fo 1) for in

A: To prove, knk=nn-1k-1

Q: Let n be a positive integer, then n+1≥ for all n.T/F

A: We need to show that n+1≥n for all n where n is a positive integer. Lets proof by contradiction

Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

A:

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

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Q: 1. Prove that if n e N, then E-1 k(k + 1) = n(n+1)(n+2)_ k=1 3

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Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

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Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: 6. Prove that for all n E w, the equality 0 n = 0 holds.

A: We will prove by induction. We need to prove for all n∈w, the equality 0·n=0 holds Let n=1, then,…

Q: For any n e N, let Bn = (n, ∞0). {B„}1 is descending. Show that Bn = 4. %3D n=1

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Q: Prove E(2k – 1) = n² for n > 1. - k=1

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

Q: Show that for all n € N, one has 1 2√n+1 <√n+1-√n<; 2√ñ

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

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Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.