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,

Q: 3) Using the expansion method, solve the following recursion and determine T(n) in terms of big-…

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Q: If TeZx Such that x is a momber of 27+5x-3=03 and S={x Such that x i. member of 2i+71+233/3 that T…

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Q: Let tn be the n-th triangular number, starting with t1 = 1, t2 = 3, t3 = 6, %3D %3D etc. (a) Prove…

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Q: Prove by mathematical induction that, for all positive integers n, de (e* sin x) = 23"e* sin(x…

A: To Prove →dndxn ex sin x = 212n ex sinx + 14nλBy mathematical induction for all positive integen…

Q: Find the value(s) of k such that A is singular.

A: The given matrix will be singular if the determinant of it is 0

Q: Prove by mathematical induction: For Vn E Z*,7|(232 – 1)

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Q: n(n + 1)(2n + 1) 2. Use induction to prove that )k2 for all n e Z+. %3D 6. k=1

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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: 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: Let T(N) = 1. T(floor (N/3)) + 1 and 7(1) = T(2) Prove by induction that T(N) ≤ log N + 1 for all N…

A: Given: TN=TfloorN3+1 T1=T2=1   To Prove: TN≤log3N+1 for all N≥1, using induction.

Q: Let G = {3"6" | m, nE Z} under multiplication. Prove that G is isomor- phic to ZOZ. Does your proof…

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Q: Exercise 4 Show that for cach k EN it holds that kn-k < n! S n", Vn E N, n2 k. Deduce from this that…

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Q: 4. Use mathematical induction to prove that Vn E Z* u {0}, []²=o(2i + 1)! > ((n + 1)!)**. n+1

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Q: 0/ the Valke of & that makes the Velteof the fumetion 2.

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Q: (b) Prove that for any positive integer n, j. 2³ = (n – 1)2"+1 +2 j=1 (c) n(n? – 1) - Prove that for…

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Q: 4. Use mathematical induction to prove that V n E Z* U {0}, [I²-o(2i + 1)! 2 ((n + 1)!)**. n+1

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Q: Let A= 3 (n) Compute det (A) using a cofactor expansion along row 2. (b) Find A adj(A). (c) Find det…

A: Result: If determinant of matrix is a than determinant of  inverse of matrix is 1a. Transpose of…

Q: 28. Prove by induction that i(i – 1) (n+1)n(n– 1) i=1 holds Vn E N,n 2 1.

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Q: 2. Find +he source (メ) Q (x) that the So lutiom U (x it) of k + Qx) 2 - 0 l• Con e

A: given equation ∂u∂t=k∂2u∂x2+Qx,       -∞&lt;x&lt;∞           ...iux,0=0 claim- to find the source Qx…

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: b) Given that z, = V3 +i and z, = -2- 3i . Find the modulus and argument of and %3D Hence, express…

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Q: Prove that eie ei(0+2K) ,where k = 0,±1, +2,...., = D

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Q: Estoblosh induction. the for mulas helow by mothemotial ncnt) a 1+213 +. th = tor all nz %3D

A: Solution: The objective is to prove the given statement using mathematical induction

Q: that y? etcos 2t and Gecos 2t knowing are solühions of. yo _uys +luy'- 20y'+ 25y=0 what is the…

A: For the solution follow the next step.

Q: Use mathematical induction to prove: m Vm, n e Z*, EEi – j) =; mn (n- m) i=1 j=1

A: Principles of induction method Consider a statement P(n), where n is a natural number. Then to…

Q: Proof: böt: if m #n 1-iħ if m= n 0, m or n = x,y,Z Pm Pn] : %3D

A: Given that, pm,n=0m≠n-ihm=n The objective is to show that, Pm,Pn=0. If m is not equal to k and n is…

Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: Let x1 V3, xn+1 = V3*" ,n > 1. Use induction to show that x, < 3 for all n E N.

A: Given x1=33,  xn+1=3xn3. Show that xn≤3 by using induction for all n∈ℕ.

Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

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Q: Express the following tunctrons in the form u+iv f(2)=In (I+z)

A: Solve the given problem.

Q: For n ∈ N, set ζn = e 2πi/n = cos(2π/n) + isin(2π/n) with i =√−1. Show that Q(ζp) is a splitting AND…

A: proof ;-  let x=e2πip  now take p power both side xp=e2πi=1xp-1=0therefore xp-1∈Qxthe root of xp-1…

Q: Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a…

A: Given dk is number of derangement of order k. We know that for a set 1,2,…,k, the derangement dk are…

Q: enoitetimil torte stigasb = 4", where m, neN. 10. Prove the identity: 3 r=0 itong s

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Q: Find all the joint moments m,1, n and k= 0, 1. . .

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Q: { (3i – 2)² = -n(6n² – 3n (b) Prove by induction that | i=1

A: We have to prove that ∑i=1n3i−22=12n6n2−3n−1 for n≥1 by using principal of mathematic induction.…

Q: Use Proof by Induction to prove that 5n+1 - 1 1+5+ 52 + 53 + …. + 5" = Vn e Z, 4 Be sure to label…

A: 1+5+52+53+...+5n=5n+1-14   ∀n∈ℤ+

Q: An Instructor teaching three sections of vering the same material. She has made ction, but she…

A: By using ANOVA test for a single factor, a) Null and alternative hypothesis: Ho:  µ1 = µ2 = µ3  (All…

Q: Find the value(s) of k such that A is singular.

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Q: 3. Prove that A=I-N+N²_N³ for A & N defined as follows: N=A–I (identity), where 1 0 0 & I=|0 1 0 1…

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Q: (c) Evaluate d Gosh: where C: |z| = 2 by using Cauchy's residue theorem.

A: Given integral is ∮C dzcoshz;   C : z=2 As, we know  If z = x+ iy then z =x2+y2 As given that  z=2…

Q: ZH E LI and HI 1 GJ. Complete the proof that AGIJ S AGHJ.

A: Here we use two concepts  1) sides opposite to equal angles of a.         triangle are equal. 2)…

Q: Apply the superposition principle to find a particular solution u = u(t, x) of the following PDE:…

A: As per the question we are given a non-homogeneous linear partial differential equation. And we have…

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: - In the domain of Gaussian integers Z[i], the element 43 is O irreducible O reducible O unit

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Q: 7. Find T, N, ar, an at t=n. r(1) = 4cos(1)i–4sin(r)j-3rk

A: Tt = r'(t)r'(t) and N(t) = T'(t)T'(t) aT = r'(t) .r''(t)r'(t) and aN = r'(t)×r''(t)r'(t)

Q: -- a) such that ab =1. Let p>1 be a prime number. Show that for each a + 0 in Zn there is b in Zp…

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.