We Section 3.1 Homework 1. 1+2+... +n' = 1n2 (n +1) for all natural numbers n. 1 +... + n(n + 1) 1 1 + 1 n 2. + for all natural num 1(2) (2)3 3(4) (n 1) (n+1) n+1 - for any r 1 and any ne N 1-r 3. Show that

Q: Show that ø(n²) = nø(n) for every n E Z*.

A: I have used only one formula of euler phi function

Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: Let n E N. Prove by mathematical induction that 3* = 4". k k=0

A: We have to prove given expression using mathematical induction.

Q: This exercise uses one or more of the following results from Section 5.2. Exercise 5.2.10: For every…

A: We will use the given information to complete the incomplete proof of given statement.

Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: Question 6.7. Prove that x'(Km) = n if n is odd and x'(Km) = n – 1 if n is even without using…

A:

Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

Q: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

Q: 2. Show that following statements are correct: i. 4n+100=0(n) iii. n³ + O(n²) v. n! = O(n) vii. 3n3…

A: Now , Tn=θfn if Tn=Ofn  and Tn=Ωfn Now, Tn=Ofn if there exist positive integers M, n0 such that Tn≤M…

Q: Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

A:

Q: Assume that S = Σ(-1)"-¹. 999 9 99 8 O 1000 10 1 (n + 1)5 n=1 n=1 Find the smallest integer N such…

A:

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 5. Use induction to prove that for every n E N we have E(k · k!) = (n + 1)! – 1 k=1

A: We want to prove ∑k=1n(k·k!) = (n+1)! - 1

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: 4. With the fact of 1² +2²+...+ n² = n(n+1)(2n + 1)/6, prove that n n n(n −1) ΣΣ(i-j)² 6 i=1 j=1…

A:

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

A:

Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

A:

Q: Prove NIn, +oo[= 0 ! n=1

A: To prove that ∩n=1∞[n,∞)=∅. Suppose that x∈∩n=1∞[n,∞). Then, x∈[1,∞), x∈[2,∞),.... That is, x≥n,…

Q: 3. Use Mathematical Induction to prove that 4 + 10 + 18 + ... + n (n + 3) = [n .. (n + 1) (n +5)] /3…

A: The solution is given as

Q: 1. Prove that if n E N, then E, k(k + 1) = n(n+1)(n+2). 3

A:

Q: 7. Prove by induction that E,(-1)² = (-1)"n(n + 1)/2 for all n > 1.

A:

Q: Prove that n2+ 4n + 6 E O(n²)

A: Solution: The objective is to prove that the given statement

Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: or every integer n ∈ ℕ, it follows that 1^2, 2^2, 3^2, 4^2 + ...+ n^2 = (n(n+1)(2n+1)/)/6.  P

A: Claim: 12+22+32+…+n2=nn+12n+16 Proof: Now k-13=k3-3k2+3k-1 Rearranging the above equation. we get,…

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

A:

Q: Prove n (2n = n n 1 k=1

A:

Q: Prove by Mathematical Induction. n(n + 1)(2n + 1) k² 6 k=1

A:

Q: Show that 2n is O(3n) but not that 3n is not O(2n)

A: We need to show that 2n is O3n but 3n is not O2n Definition of Big O notation: Let f be a real…

Q: 1. Prove by induction that for all positive integers n, we have: 1+3+5+ + (2n 1) n2

A:

Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

A:

Q: 1. Prove that {2" : n E Z} = {,n :n E Z}.

A:

Q: Verify by mathematical induction Ž(2i-1 1. 1 - n° (2n? – 1) for all n 2 1. n(n+1) 2. Prove: V n € Z,

A: As per our guidelines, we are allowed to answer only one question. Since you have asked multiple…

Q: Prove that (n+1)C n is equal to n+1

A:

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

A:

Q: 1. Prove that if n e N, then E-1 k(k + 1) = n(n+1)(n+2)_ k=1 3

A:

Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…

Q: Show that if M is a number less than 1, then the terms of {n/(n + 1)} eventually exceed M. That is,…

A:

Q: Let T = > Σ - In k. n n=1

A: We will solve here only (a) part of 2.   we have to show that the sequence Tk is monotone…

Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

A:

Q: 6. Prove o (k2) = k · Þ(k) for all k E N.

A:

Q: To prove 3+7+11+...+(4n − 1) = n(2n + 1) holds for an arbitrary n € Z+, in the inductive step, under…

A:

Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: Choose the answer that best completes the proof that inf(A)=0 where A = for all nEN n? Firstly, we…

A: Given the sequence A=1n2:n∈ℕ.  We need to prove that infA=0. The elements in the  sequence is of the…

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.