If o is an imaginary cube root of unity ,Prove 1 1 1 1+20 2+ @ 1+@

Q: Show that the square roots of two successive natural numbers greater than N2 differ by less than…

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Q: Prove the square of any integer has the form 4k or 4k+1 for some integer k

A: Consider the set of all integers , To prove that the square of any integer is of the form 4k or 4k+1…

Q: Check the definiteness of the following quadratic form; Q(x1, X2, X3) = 2xỉ + 3x – xỉ + 5x,x3 + x,X3

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Q: If z1 and z2 are complex numbers such that |21| = |2|, prove that arg(2122) = arg ((21 + z2)²) .

A: This  problem is related with complex analysis. Given that z1 and z2 are two complex numbers such…

Q: Let a EC be a primitive nth root of unity.

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Q: Show by induction: If r +0, then 1- 1+1 Σ 1-1 k=0

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Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: For any n >1, prove that the sum of all the nth roots of unity is 0.

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Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: 1. Prove that the square root of 2 is not a rational number.

A: Definition of rational number : If x is a rational number then it can be written in the form of pq…

Q: Prove the square of any odd integer is expressible in the form 8n+1 with n ∈Z.

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Q: If a, b, c are integer powers of 2, prove that equation 3 a° + bª = c5

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Q: Which of the following is a 4th root of -1.25-12.3? 0.9931 +j 0.8113 O 1.4454-j 0.4312 -0.5569-j…

A: The given problem is to find the fourth root of the given complex expression. Given complex…

Q: If a, B are the roots of the equation x + px + q =0 and also of x + p"x" + q* = 0 %3! αβ and If are…

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Q: by p. 13, and 17. 2 For n= 1, 2,..., 10, show that 5u +4(-1)" is always a perfect square. 3. Prove…

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: 1 1 prove that 1+2@ 2+W 1+w If w is an imaginary cube root of unity,

A: If ω is an imaginary cube root of unity then (1) 1+ω+ω2=0(2) ω3=1

Q: (b) Let A1, A2, · ·, A55 be the roots of 81/55, Show that 55 d; =0

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Q: Find the 2 square roots of 3 - 4j using De Moivre's Therem.

A: Let z=3-4j---(1) By De Moivre's theorem, z=r cos θ + j sin θ ⇒z=rcos θ + j sin θ12 ⇒z=rcos θ2+j sin…

Q: 1. Give the ten 10th roots of unity. Write them in the form e. Among these 1Oth roots, which are the…

A: According to Bartleby's guidelines, we are allowed to answer only one question at a time. So here is…

Q: Check the definiteness of the following quadratic form; Q(x1, X2, X3) = 2xỉ + 3x – x + 5x,x3 + X2X3

A: The definiteness of quadratic form is defined by corresponding symmetric matrix.

Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: (10') Use Rouche's theorem to find the number of roots of 2z4 – 5z + 1 = 0 (1) for |z| < 1 (2) for…

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Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: Calculate all the primitive roots of 41 and 26.

A: (a) Consider a=41. Therefore: φa=φ41=411-410=40 Now φa=40 can be written as: φa=40=23×5. Therefore…

Q: a) Calculate the 4th roots of (5-j3)

A: The objective is to calculate the 4th roots of (5-j 3).

Q: Show that if u is an odd integer ,then the equation x2 +x−u=0 has no solution that is an integer.

A: The objective is to prove that x2+x-u=0 has no solution that is an integer. Suppose that there…

Q: a. If b is any nonzero real number, prove that |1/b| = 1/| b | . b. Prove that |a /b | = |a|/| b|…

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Q: prove that 3^1/2 is irrational

A: We have to prove that 3^1/2 is an irrational number.Let us assume that 3^1/2 is a rational number.…

Q: Prove: The difference of the reciprocals of two successive integers equals the product of their…

A: Given, 1n-1n+1    We have to prove the difference of the reciprocals of two successive integers…

Q: When a is a positive real number, the ________ ________ root of −a is defined as √−a = √ai.

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Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: Which of the following is a cube root of – 2.3 – j1.25 1.3782 + j3.3075 -1.3593 - jo.2276 -1.3782 -…

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Q: Prove that the roots of ax+4=2x²-bx are real for all values of a and b

A: The nature of roots of a quadratic equation Ax2+Bx+C=0  is:

Q: (b) Let A₁, A2, · · ·, №46 be the roots of 2¹ 1/46 that 46 Σλ = 0 i=1 . Show

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Q: find all real zeros for the given polyn 20x + 12; x + 3

A: Polynomial: f(x)=3x3+x2-20x+12 The zeros means at those values f(x) will be zero.

Q: if n is a perfect square then n2 + 2n is not a perfect square. prove this

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Q: Prove the given statement: The sum of the squares of two odd integers cannot be a perfect square.

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Q: QUESTION 5 If x and y are two incongruent primtive roots of a prime p. is x+y a primitive root of p?…

A: In this problem we shall use the definition of primitive roots to show if the number given in the…

Q: Question is attached, Thankyou

A: Let us take z = x + iy.

Q: 2. Solve for the complex mumber z in r (4+ 36) – 7 = 7x - 4zi. Show all algebra.

A: Given expression is x(4 + 3i) - 7 = 7x - 4xi

Q: Prove (NOT BY EXAMPLE) that for every nonzero integer, its square is positive.

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Q: a) Compute the four 4th roots of −T + Ti

A: Compute the fourth roots of z=-T+Ti, where T is any arbitrary real number.

Q: Identify the real fourth root(s) of 256. O -4 O -2 O -1 O 2 O 4 O 2 and –2 O 3 and -3 O 4 and -4 no…

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Q: Let p and q be primes. Prove that pq +1 is a square if and only if p and q are twin primes.

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Q: Prove that the product of all the distinct nth roots of 1 is ( - l)n+ 1

A: Let us complex number zz=(1)1n apply power n on both sidezn=1zn-1=0

Q: Show that 2 is a primitive root of 19.

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Q: Use DeMoivre's Theorem to find the following. Write your answer in standard form. (1 + 3) - 16/3 +…

A: DeMoiver theorem state that : {r(cos(theta) + isin(theta))}^n = r^n(cos(n*theta) + isin(n*theta))

Q: Ф(тk) — ф(0) — тk

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.