Use mathematical induction to show the following. n 2(√n +1 − 1) < Σ √√/i Ë < ² v < 2√n +1 i=1

Q: Let = e²i/n be an n-th root of unity. Show that (k(1-n) Resck (27²1) n for all 0 <k<n-1. =

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Q: Find the first three terms of the expansion (4 – 2x)³.

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Q: P(X₁ = x₁, ..., Xn = xn) = P(X₁ = x₁) [I²²2 P(X; = Xi| X₁ = X₁, ..., Xi−1 = Xi−1 )

A: Here we have a given equation that

Q: Find the smallest integer x for which (x4 + 4x)/(x + 2) is O(xx). 01 02

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Q: Evaluate O x3+4x²+ 6x Select the correct answer below: -1 2² ) ) + 0 O (2 In [x]-In (x² + 4x+6) +…

A: Given thattaking firstly using the partial fraction decompositionon comparing the like terms

Q: Let n be a positive integer. Find the continued fraction expansion for √n(n + 1).

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Q: n₁ Evaluate the following formula for p₁ - P₂ = 0, x₁ = 64, x₂ = 11, n₁ =311, n₂ = 296, p₁…

A: Given: P1-P2 = 0, X1 = 64, X2 = 11, n1=311, n2 = 296

Q: n √n II k=1 lim √n n1x 2k 2k + 1 = ?

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Q: The height of a ball is modcled by the function h(t) seconds. What are the highest and lowest points…

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Q: The Lucas numbers can be generated using the recursive formula: L1 = 2 L2 = 1 LN =LN–1 +LN–2 What is…

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Q: (4 – 2æ)".

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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: Transform the following sum by making the change of variable j =i-3. Here, n23 is an integer. 2 – 4i…

A: We use the given relationship between i and j to transform the given sum.

Q: Find the first three terms of the expansion (4 – 2x). 4°(1- 폴 + 쫄 +…) O b. 2°(1 + 꼭 + 플 + …) Oc…

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Q: Give a proof by cases that [4x] = [x] + [x + ¼ ]+ [x + ½ ] + [x + ¾ ] %3D

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Q: Find the term of (a – 2b)3 that has b in it. |

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Q: What is a simpler form of 5n(3n3-n2 + 8)?

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Q: Find the minimum and maximum values of the function f(x, y, z) = x² + y² + z² subject to the…

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Q: Solve for t in B=Numerator x+y over the denominator t+z

A: The given equation is,

Q: (b) Prove that 32n²+1, for n ≥ 1.

A: The given inequality is 3n≥2n2+1, for n≥1. We have to prove this inequality.

Q: FOR N=195, by fermat's theorem an-2 =

A: Given: FOR N=195, by Fermat's theorem an-2

Q: Show that (m) (2n 4n n

A: As n → ∞, By Striling approximation formula n! tends to  sqrt(2 pie n) (n/e)^n

Q: Let N (h) be the approximation of f'(xo) with some numerical differentiation scheme depending on h.…

A: To find the higher order approximation, use the Richardson's extrapolation formula for…

Q: Evaluate f O 3√x+1 2 (x+1)² 2 O 3(x+1)² 2 dx 1+ 3√x+1 +3√x+1+C +√x+1 − 3 ln|√x+1+1| + C + 3√x + 1 +…

A: Given, ∫11+x+13dx

Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: Find the seventh term of the 10 expansion ( x+ y)º

A: We know that (x+y)n = ∑r=0nCrn xn-r yr Hence (x+y)10 = ∑r=010Cr10 x10-r yr

Q: For positive integer n, prove the following (given that I() = √T): T(n + 1) = 1-3.5 (2n-1)√√ 2" =…

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Q: 2. 2 +3+5t t(2n-1)=h(2n-1s (2nt) %3D 3. for all nz I

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