4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ + 23 + ... +n³) – (1³ + 23 + ...+ (n – 1)*).]

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

Q: Prove: For integers m, n, 12mn – 9 is an odd integer.

A:

Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

A:

Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

Q: 3. Let n and m be positive integers. Evaluate > (km+1 (k – 1)m+1) if possible. If not possible, k=1…

A: We have to evaluate the given summation.

Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: Prove that 2" < n! for any integern 2 4

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

Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

A:

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

A:

Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

A:

Q: Show that 3n^ 2+8nlogn is big-Oh (n ^2 )

A:

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

A:

Q: The general combinatorial formula for the nth triangular number (1, 3, 6, 10, 15, 21, .) is On+1Cn…

A:

Q: Prove that 3 is irrational number.

A: Rational Number: A number is of the form of pq, where p, q∈ℝ and gcd(p, q) = 1 Now we have to show…

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

A:

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: 11. Prove that if n is a perfect square, then n + 2 is not a perfect square.

A:

Q: 1. Show that the distance between two numbers in the real number line, given by d(x,y)=|x-yl,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1. Prove that V3 is irrational number.

A:

Q: Prove there are no natural numbers n and p with n 2 2 and p² + 1 = 2".

A: prove by contradiction:- to prove it by contradiction, you should start by assuming its negation,…

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: Prove the following using a direct proof: The sum of the squares of 4 consecutive integers is an…

A: To prove that the sum of the square of 4 consecutive integers is an even integer.

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: Compute the infimum and supremum of the set {n+(1/n)" : n E N} if they exist.

A:

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

A:

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: 2. Find all Pythagorcan Triples (not just primitive ones), containing the integer 15.

A: The Pythagorean triples are positive integers a,b,c such that a2+b2=c2. We can say that the…

Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

A:

Q: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

A:

Q: Prove in a thoroughly written proof that for all integers a  and m, if a and m are the lengths of…

A: Recall the following. Pythagoras theorem: For any right triangle, the square of the hypotenuse is…

Q: II. Prove the following using Direct Proof. 6. If x is an odd number and y is even number, then x +…

A:

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: Prove there are no natural numbers n and p with n > 2 and p² + 1 = 2".

A: ANSWER:

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: Prove that: there exist infinitely many 5-tuples of practical numbers of the form (n-8, n – 4, n,…

A: It is given that 5-tuples of practical number of the form (n-8, n-4, n, n+4, n+8)

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: . Prove that there are no integers n with o(n) = n/13 %3D

A:

Q: Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In…

A: Introduction: Divisibility is a property of an integer. For two integers a and b with b&gt;a, the…

Q: 2 There are integers m and n such that m² +n² is a perfect square.

A:

Q: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…

Q: prove that, it ņ is an 3n+2is oald then integer and nis add

A:

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.