The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1, whit x1 = 1.Prove that (xn, xn+1) = 1 and that xn (a" – b") /(a – b), where a and bare the roots of the quadratic equation x? – x – 1 = 0. X2 - -

Q: Prove that x4+4=(x+1+i)(x+1-i)(x-1-i)

A:

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: Suppose the conjugate z is equal to the reciprocal 1/ z. What are all possible z's?

A:

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

A:

Q: X sec10 can 1. Let u = 7 In х2 - 1

A: Given: ∫x sec107ln2x2 - 13tan7ln2x2 - 13dxx2 - 1.

Q: Find the number of ideals in Cx,y that contains both the polynomials y – (x – 1)³ +1 and p(x) in…

A: We have to find the number of ideals in ℂx,y that contains both the polynomials y-x-13+1 and px in…

Q: 5. Use De Moivre's Theorem to solve all the roots of x^5 +1= 0 O A. 1 cjs [(nt+2nk)/5] ; k = 0, 1,…

A: Given that x5+1=0. We know that  x+iy=r(cosθ+isinθ(x+iy)n=rn(cosθ+isinθ)n =rn(cosnθ+isinnθ)…

Q: 2.6. In the text, the modulus is defined in terms of the inner product. Prove at this can be turned…

A:

Q: 2. Find all number z such that Az" + Bz² + Cz = D a. A= 2

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

Q: Which of Zs, Z2o are cyclic? 18, '20

A:

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…

A:

Q: Suppose x,y,z are mutually coprime positive integers that solve x2+y2=z2. a) Show why exactly one…

A: Basic properties of modular arithmetic and gcd will be used in solving this problem. The detailed…

Q: Prove that (3-2)(x+2) = 3+2i ifitues.

A:

Q: 1 Expand in ascending powers of x, up to and including the term in x', simplifying the V(1 –2x)…

A: To solve the following problem

Q: Q4: 1- Prove that (1-0+(1+ () 0. 2. Put in the form a+ib ((1+i))- ((1-i)).

A: Here we have to prove that, 1) 1-i2+(1+i)2=0 2) Put in the form of a+ib, 1+i22-1-i22

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: Find a number t such that In(4t + 3) = 5 %3D

A:

Q: Prove that the equation x+5x=4cosx cannot have more than one real ro

A:

Q: The th roots ofz=a+ bi whenk=0, 1, ..., n-1. take the form: 0+2n COS 0+2n +i sin O [cos(0+ 2kx) +i…

A:

Q: Find an explicit formula for a sequence ao, a1, a2, a3, . . . that satisfies — 24. (Hint: ат Cr" +…

A: To find the explicit formula for a sequence given using recurrence relation find the characteristic…

Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

A:

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

A:

Q: f for ƒ : R - {2} → R- { 5} indicated by f (x) =(5x +1)/(x -2a), a bijection can be made from [0, 1)…

A: Bijective function: The domain is set A and the co-domain in set B relates elements of two sets A…

Q: Find {2), f3), R4), and {5) if fis defined recursively by f0) = -1, A1) = 2, and {(+1) = Jtn}*s(n –…

A:

Q: 19. Find an interval of length i in [0, 1] containing a root of x - 5x +1 = 0.

A:

Q: Proceed as in Example 3 in Section 6.1 to rewrite the głven expression using a single power serles…

A: Given ∑n=2∞nn−1cnxn−2+∑n=0∞cnxn+2 We have to rewrite the given expression using a single power…

Q: 4. Prove C(n, r) = C(n- 1,r- 1) +C(n- 1, r) %3D using a combinatorial proof.

A: We know, C(n,r) = n!/((n-r)!r!)

Q: state and prove the Cauchy's Residue Theorem?

A:

Q: Let Polynamial. has infinitely mny Toats, R-ZO7. Show t the (1,0)xin roots, Sol

A:

Q: (i) find gcd(2,3+5i) in Z[i], (ii) show that 3+4i and 4-3i are associates in Z[i] , do all , dont…

A: Solution (i)

Q: (1) Use the definition to prove that if xn → 2 then (2xn + 1) / xn → 5/2.

A: As per our company guideline we are supposed to answer only first three  qs, kindly post remaining…

Q: Prove that In( +VI-X²³) = In ( ¹+ √1-x²) sech ¹(x) =

A:

Q: 8. Let B = and A = Verify that AB, the pseudoinverse of AB, is not equal to BA. Thus the formula…

A:

Q: Theorem 4.8.1 Irrationality of V2 V2 is irrational.

A: For any integer n such that n² is even then n is also even.

Q: an 34an-1 – 289an-2 for n > 3 with initial conditions aj = 68, az = -1734. an

A:

Q: 4. Sketch the proph of the following funcfions; メーメ+4 (x+4 a) f(x) = (x+4) x b) gex) = %3D

A:

Q: Question is attached, Thankyou

A: Let us take z = x + iy.

Q: Find a number z that satisfies: |z| - z = 2 + i

A: Complex number which satisfy |z|-z= 2+i

Q: Show that there exists root in the interval [1, 2] of x = 2sin(x), and then use false position…

A: Intermediate value theorem:- If f is a function which is continuous at every point of interval a,b…

Q: Given two polynomials in from of x^3+x^2+x+c =0 x^3+x^2+x+c =0

A: Given: To find their intersection using matrices. Given two polynomials in from of…

Q: How can I find all monic divisors of degree 1 in Z_7 [x], 5x^2+5x+4? please help me get started with…

A: Set of all Polynomials over a field ℤ7 is given by ℤ7x=anxn+an-1xn-1+ . . . +a1x+a0  where a0,a1, .…

Q: 6. Use the IVT to prove that x2+ In r = has a solution. %3D

A: The Intermediate value theorem says: If the continuous function attains two values then it must…

Q: #1 and #2, write a 2-column proof. Given: 5(2x - 8) = 30; Prove: x = 7 %3D 5C2x-8)3D30 Gioen

A: topic:-simplification

Q: Find a polynomlal f (x) of degree 3 that has the following zeros. 5, -7, 6 Leave your answer in…

A:

Q: Transform of the followi 2. + (5) = 35-25-1 (S-3) (5²+1)

A: Given that  F(s)=(3s^2-2s-1)/((s-3)(s^2+1)) To find the inverse laplace of F(s)

Q: E Show that x'+ 3x+ 2 has 4 zerosin Ze

A:

Q: state and prove the Cauchy Residue theorem?

A:

Q: (a) State the Eisenstein Criterion (without proof). [ (b) Prove that 30x³ 91 does not have roots in…

A: