1.4. Problem 4. We say that a E R is badly approximable if there exists a positive constant c> 0 such that for all rational numbers p/q e Q We proved in class that 2 is such a number. Prove or disprove: if a ER and B ER are both badly approximable, then a + B is also badly approximable.

Q: Show me the steps of determine red and the inf is here

A: Given:                                xn+1=Axn+Bxn-k+Cxn-l+Dxn-σ+bxn-kdxn-k-exn-l, n=0,1,⋯ ...(1) To…

Q: Question 5. Consider the following three statements about a real number x: • P(x): -1 ≤ x ≤ 3. •…

A: P(x) : -1≤x≤3 Q(x) : x2 > x  x2 > xx2 - x > 0x(x-1) > 0x > 0 and x - 1 >0 ⇒ x >…

Q: a) The Basel problem asks for the precise summation of the reciprocals of the squares of positive…

A:

Q: e b (A+C+D) (38) PQ= (e+d) [(1– B) +K3] [d (1 – B) – e K3]²

A:

Q: PROBLEMS 2.5 1. Which of the following Diophantine equations cannot be solved? (a) 6x + 51y = 22.…

A: By finding GCD and using extended Euclidean algorithm you can solve the problems as follows :

Q: 1.(a) Do there exist integers a, b, c such that gcd(a, b) = 30, gcd(b, c) = 24, gcd(a, c) = 54?…

A: (a) gcd(a, b) = 30       gcd(b, c) = 24       gcd(a, c) = 54      => 30 | a, 54 | a           24…

Q: Question 4. Prove that for any x, y ≥ 0, we have |√x - √y| ≤ √√√x – y\. (Hint. Consider the cases…

A:

Q: ELEMENTARY NUMBER THEORY 9(2n + 1) = t9n+4 – t3n+1 -

A: Triangular numbers: The number that computes the things set up in an equilateral form is called the…

Q: iestion 3. Prove that for any numbers x and y with r ‡ 0 we have 2y ≤ ½ + x².

A: Answer given below.

Q: 4. Let bo = 1, b1 = 3, and for cach integer k> 2, we have bz = 4bg–1 - 4bg-2 Prove that for all…

A:

Q: (4)

A: Solution-

Q: Let a and b be positive integers with ged(a, b) = 1. Show that if d > ab, then as + tb = d has a…

A: 4. Given that a and b be positive integers with gcd(a, b)=1.     To prove that if then has a…

Q: 5.72. Evaluate the proposed proof of the following result. Result If x is an irrational number and y…

A:

Q: Write a piecewise function for each function graphed below. 12. 13. 14. Gina Wilson (All Things…

A:

Q: Problem 10.1.2. Let (xn), (yn) be sequences as in the NIP. Show that for all п, т € N, zn < Ут-

A: see attached image

Q: Problem 21. Prove that a real number y is computable if and only if there is a computable function f…

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Q: 2. Let f(x) = -x– 1. a) Find f-1(x).

A:

Q: 5 = t(n + 1) = t(n +2) = t(n + 3) holds for n = 3655 and 4503. (a) Verify that t (n) (b) When n =…

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Q: - 21. (a) Prove that if d | n, then 2d -1 | 2″ – 1. [Hint: Use the identity k-2 x − 1 = (x − 1)(x*−1…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.