State and prove the De Moivere's Theorem for the following cases (i) n is +ve integer (ii) n is -ve integer (iii) n=o (iv) n is rational number

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: (5) 7(3k – 1)² +26(3k – 1)–8

A: On simplifying the above expression,we get

Q: (b) 98

A: The given number is 298. A number is called a rational number if it can be represented in the form…

Q: Consider all positive integers with 3 different digits. (Note that 0 cannot be the first digit.) (1)…

A: Iam assuming that the numbers from 0 to 99 are not valid because they are not 3 digit numbers, so it…

Q: For the following proof, determine which of the statements given below is being proved. Proof.…

A: Contrapositive: The contrapositive of the statement, " If P, then Q" is " If not Q, then not P".

Q: Suppose a and b are positive real numbers and n is a natural number with n 2 2. (a) Use the…

A: Let a and b are positive real numbers and n is a natural number with n≥2. Use the difference of…

Q: Prove that 2n + 3n is a multiple of 5 for all odd n € N.

A:

Q: The following statement is true. V nonzero number x, 3 a real number y such that xy = 1. For each x…

A:

Q: Find the sum 1 9k2 + 12k – 5 k=1 Express your answer as a rational expression in terms of n.

A:

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: ind the interval in the real number line which equals 1 U [4+, 20- n n=1 n

A: Answer given below.

Q: If a is a real number and n ≥ 2 is an integer, then which of the following expressions is equivalent…

A: Given expression: an

Q: 2. Prove: Proposition: If n E N, then 1 + (-1)^(2n – 1) is a multiple of 4. 3. Prove: Proposition.…

A: We use mathematical induction concept in 1st question

Q: Prove the following statement directly from the definition of rational number. The difference of any…

A:

Q: 6. For an arbitrary integer a, verify the following: (a) 2 a(a + 1), and 3 a(a + 1)(a + 2). 4(b)…

A: a is an arbitrary integer.

Q: For each of the statements below (which all happen to be false), look at the “proofs” that have been…

A: Statement: The sum of any three odd integers equals 3p for some integer p.In the proof of this…

Q: Every nonzero rational number can be written as the product of four irrational numbers.

A: Let x be any irrational number       Step(1):  First we will prove…

Q: In 1742, Christian Goldbach conjectured that every even number greater than 2 can be written as the…

A:

Q: show that there does ot exist a rational numbers such that s2 = 6

A:

Q: Express all real numbers greater than 6 and less than or equal to 7 in interval notation.

A:

Q: Please prove this question the other way around...   10. Is the following proposition true or…

A: We have to show that  is even iff and only iff  divides

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.