PROBLEMS 1.1 1. Establish the formulas below by mathematical induction: n(n + 1) (a) 1+2+3+… +n = for all n > 1. (b) 1+3+5+...+ (2n – 1) = n² for all n > 1. n(n + 1)(n + 2) (c) 1- 2+ 2 · 3 + 3.4+ . ..+ n(n + 1) = for all n > 1. %3D

Q: in is where n is an integer. equal to the above e(1+2n)i wheren is an integer. equal to the above…

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Q: anced Problems 1. Evaluate using ratio test. 11 1 1. 1 n! n+1! 1+1++. 2! 3! 4! ... a. (expand the…

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Q: In2 + 5n + 4 lim an = 1 %3D %3D 3n2 + 8n + 8 n-00 1n? In2 5n + 4

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Q: Given: In(1+ x) (-1)" xn+1 _if |x| < 1 n + 1 n=0 (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of…

A: Solving

Q: (n^2-n)+((n-1)^2-(n-1))+....+(1^2-1) =n(n+1)(2n+1)/6 - n(n+1)/2

A: We shall first derive the formulas for sum of first n natural numbers and sum of squares of first n…

Q: Reading Section 5.1 Problems Prove (by induction): For all n E N: 21 |(4"+1 + 52n-1) Make sure that…

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Q: QUESTION 4 The Generating function for the sequences { k) with k = 3/2 o a)1 + + 2+ 3+ 4, b)3 + 3…

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Q: Question no. 42 Suppose that [1/4 1/2 1/4 P= a 0 1-a

A: The states are {0,1,2} P=0.250.50.25α01-αβ01-β Pijn=P(Xn=j|X0=i)piin = prob of return to i in n…

Q: Problem 14 What is the coefficient of x101ly" in the expansion of (2x – 3y)²00. What is the…

A: Given that  The following problems 14  To find the coefficient

Q: A Moving to another question will save th Question 2 E K equals to k=1 O a. (n+1)(2n+1) 6. O b.…

A: We can proof by series sum formulas

Q: the coefficient of x^2n in the expansion of (1-x)^2n is

A: Hello, thanks for your question but according to our policy I am answering the very first question.…

Q: (-2)"+1 n+ 5" (2n)! 2n - n! - n (a) (b) (-1)". n=1 n=1 (c) E(-1)": (n!)'3" (2n + 1)! (d) 5 (-1)"…

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Q: Question 26 . An infectious disease has a reproductive ratio Ro =2.8. Infectious individuals remain…

A: Given, the reproductive ratio of an infectious disease is R0=2.8. During the infection period, it…

Q: 2. Find the formula of the general n th term . (i) (1,3,5,7,.) (1 2 3 4 (ii) 2 3 45" 11 (iii) 48 16'…

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

Q: Problem 2: Show that (n + 23)³ is O(n³)

A: f(n)∈O(g(n)) iff ∃ scalars C,n0 such that f(n)≤Cg(n) ∀n≥n0

Q: The area to the right of z =0.99 is Part 2 of 4 The area to the right of z = -1.78 is . Part 3 of 4…

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Q: Given: In(1+x) = >) (-1)". -xn+1 if |x| < 1. n+1 n=0 Compute the exact value of -1)"(n+2) 2n+1(n +…

A: Given: ln(1+x)=∑n=0∞(-1)nn+1xn+1 where x&lt;1To Compute: Exact value of ∑n=0∞(-1)n(n+2)2n+1(n+1)

Q: 4. E=1(-1)k In k

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Q: Question 3. Show that n3 + 4n2 +1 O(n²). %3D n+3

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Q: (Зп+5 Find infimum of A |n E N} n+9

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Q: Problem 2 in the picture

A: We need to prove that for each n∈ℕo (set of all natural numbers),

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Q: For every n E N, 1+3+5+ ·.+ (2n – 1) = n² + 3.

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Q: а) nyn n=1 Σ >(-1)n+1 п c) 2" n=1 00 e) 2(-1)n+1, en n=1

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Q: n-1 n- n-1 n-1 () %3D h - 1 h-p n-1 n-1 P-h = q h-p n-1 ()* (1 -4) n-1 %3D n-1 h - %3D

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Q: 1.6. Problem 6. Use Induction to prove that for all n E N21 E(1+ sin (i)) > 20 i=1

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Q: Question 7 What is the sum? n+5 n2 +3n - 10 3n2 3n2 + 5n- 10 3n2(n-2) B. 23n2 +35n +50 3n2(n-2)(n…

A: sum=?

Q: If (ar + bx) 8n and n+1 k=1 n Σ (α- br) 1 1. then ar n | n+2 k=1 k=1 O 2 None 9. 2 5 2 3 2

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Q: - Problem 1 Prove that induction proofs actually works. Mathematically, |(P(0) ^ (P(k) = P(k+ 1))) »…

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Q: (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of n=0

A: The ratio test can be used to assess if a series is converging or diverging. The following are the…

Q: Provide NEAT and COMPLETE solutions for the following problems. 4. Let n ∈ Z. Prove that if n is…

A: Given: Prove that if n is odd, then n = 4k + 1 or n = 4k + 3 for someinteger k.

Q: Problem 7.3.4. Show that (n)=1 diverges to infinity.

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Q: n=1 B. E(-1)"- In n n=2 1 Inn c.(-1)"+1. n- Inn (-1)*+1. n + n=1 E.(-1)" sin(n) n=1 E(-1)"- n=2…

A: This is a problem of series calculation. based on the several available tests we can determine which…

Q: Question 9 Is 2n² + 12n + 18 equivalent to (2n + 6) (n + 3)? True O False Question 10 Is 12n + 18…

A: Note - ( As you asked multiple questions , According to our guidelines we will answer first three…

Q: The following series 3 2n+1] (-1)" 4n 3n n=1

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Q: Problem 5.3 Prove that n n(n + 1)(2n + 1) (3n² + 3n – 1) - Σ- 30 j=1

A: We have to prove given result by Principle of Mathematical Induction.

Q: Question 67 Find the sum (4z2 - 5z) + (2z° - z2 + 3z) A. 6z3 - 6z B. 6z - z2 - 2z C. 2z3 + 3z2 - 8z…

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Q: 5) 1+ + + 2/2' 3/3 8 5V5 n² + 12 6) E 3n2 + 4n – 1 n=1

A: As per our guidelines, I can answer maximum one question. Since you have asked for multiple…

Q: QUESTION 3 Frank is inviting six friends to dinner, He has 30 ciose friends. 20 are woman and 10 are…

A: (3) Given Data: The total number of close friends is: n=30 The number of woman close friends is:…

Q: Question 5 The series 2 31-2n4n+1 n =1 a can be rewritten as 2 ar". Then n =1 7 is? twith Camsea

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Q: Question 10 When fully multiplied out, what is the 8th term of (r – 3y)'3?

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Q: Question 4 3 k equals to ,2 n(n+1) O n(n+1) o n(n+1) (2n+1) 6. Moving to

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Q: [1 1 1 If A = 1 1 3"-1 3"-1 3"-1 prove that A" =| 3"-1 %3D 1 1 3"-1 for 1 3"-1 all values of n e N.…

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Q: 1. The fourth term in the an = 2n^2 – n^3

A: Dear student as per guidelines we answer the first question only when multiple different question…

Q: Find the sum of the following: 1 Σ 5n² +5n n=1 a. b. n=1 3+2-1 4n

A: a) ∑n=1∞15n2+5nb) ∑n=1∞3+2n-14n series converges: The series  ∑n=1∞an converges to k, then the…

Q: QUESTION 2 For a GP, 1/2, 1/4, 1/8, … the sum of the first term S1 = 1/2, the sum of the first two…

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

Q: Problem 2: (a) Use mathematical induction to prove that 3- 5" >n. 3" + 4"+1 for all integers n > 3.…

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Q: Problem 27: Show that N when z = 1 X (z) = 1+z-+z-2 +...+z-(N-1) 1-z-N %3D when z + 1 1- z

A: We have to prove that the sum of the given series is equal to RHS.

Q: Problem 5.26. The Fibonacci numbers Fo, F1, F2,... are defined as follows: if n = 0, if n = 1, Fn-1…

A: We will prove the given statement.

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.