Using Ordinary Generating Functions prove in how many ways the sum of 5 dices give n as a resul

Q: Chapter 4: 4.6.8, 4.6.9 (a), (b), 4.6.11 (b), (c), 4.6.14 1. Prove that for any natural n the…

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Q: Double Tower of Hanoi contains 2n disks of n different sizes, with two disks of each size. You must…

A: To find the solution, use the closed-form solution to the Tower of Hanoi. The minimum number of…

Q: (4) Determine the generating function for the sequence ho, h1,..., where hn = #((a1, a2, a3, a4) |…

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Q: 7. Use induction to prove Bernoulli's inequality: If 1 + x>0, then (1+ x)" al+ nx for all natural…

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Q: Using the generating function to determine the number of ways  18 identical balloons can be given…

A: Given that 18 identical balloons can be given to 3 children such that each child receives at least 5…

Q: Determine the sequence of the generating function A(s) = (1– 2 -9)2 |s| < 1

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Q: Let (s) be a sequence of real number, we say s, is Decreasing if for each nEN Sn >Sn+1

A: Given a statement

Q: 5. a. Find the binomial generating function for the sequence a = 1, 3,9, .....,r + 1, .. b Given the…

A: Sequences: Sequence is a list of numbers or objects that placed at a special order. Relations: A…

Q: (c) Prove by induction that 3n5n² for all n € N with n ≥ 4. State clearly where you use the…

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Q: 1 Answer: diverges n In n n=2

A: Given: S=∑n=2∞1nlnn. To be find: given series converges or diverges. Concept: for the series…

Q: 7" is 3n+1x5 Jnz1 The sequence a. geometric O b. diverges O C. Bounded O d. arithmetic

A: Geometric series

Q: Using mathematical induction. Prove that Ž {i= n(n+1) WHEN 12 1 2

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Q: 12. Prove the following facts about the Fibonacci sequence (Fn): (1) (Chapter 10, #26) E-1 F =…

A: Given, The Fibonacci sequence (Fn) Fn=Fn-1+Fn-2, where F0=0 &amp; F1=1

Q: 3 4 N + 1 5) Show that the sequence 7 2N + 1 is bounded and monotonic.

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Q: (i) Give a definition of Fermat's Little Theorem. (ii) For all n e Z, show that 30| (n³ – n).

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Q: Let a, be the number of strings of symbols {a, b, c, 1, 2, 3} of length n, such that exactly one of…

A: Given: Only one of the character is a numeric digit from the set(1,2,3,a,b,c).

Q: 4. For the sequences L (X1, X2, x3) = (50, 40, 10) %3D and (y1, y2. y3) = (3, 1,5) %3D 3 The max of…

A: Given:x1,x2,x3= 50,40,10and y1,y2,y3=3,1,5This means x1,=50 x2,=40 x3=10and y1,3 y2,=1…

Q: 7. The famous Rodrigues formula states that H, (x) = (-1)"e*² (4)" functions for positive integers…

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Q: A Sequence is a whose domain is natural number and rang is real ........... unbounded .a O function…

A: Sequence: A sequence of real numbers is defined as a function f:N→R. Here N is the set of natural…

Q: nk - k > 2 + 1' converges. n=1 Select one: O True O False

A: We will use limit comparison test

Q: 1 diverges by comparing with n-. 16. Show that n² – 3 n=2 n=2

A: Given: The series ∑n=2∞1n2-3 To show: The given series converges by comparing with ∑n=2∞n-1.

Q: a) What is the maximum possible order of an element in S,? Give an example of one of these elements.…

A: Given, Group S7 any element can be written in cycle notation i.e. in disjoint cycles as example…

Q: A sequence {an} is called increasing if an < an+1 for all n E N. ylgn ihi

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Q: 6:11 4 Search Vn +2- Vn +1 Vn + 1yn +2 n=2

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Q: the can be generated by f(n) = {Fn – 1) + if n = 0, The sequence is O 1,2,3,4,5,6,.. 1,3,5,7,9,.. O…

A: Given that f(n) =  f(n-1) + 1 ; n&gt;=1 And f(0) = 0

Q: (d) Investigate some other examples. Is there some fairly simple test to determine whether a given…

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Q: 6. Given a sequence ro, z1,2,... which satisfies the relation z,= 20 = 0, a1 = 2. Let 4zn-1- 3zm-2…

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Q: , How many ways are there to distribute r+s identical balls among five different children such that…

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Q: Q\ Is the sequence (Vn + 11 – vn+1)y bounded ? nEN Neither Yes No

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Q: Exercise 8: Let x ER\Z, and suppose x has a finite number of decimal places. Let n be the numbe of…

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Q: A sequence {an} is called decreasing if an < an+1 for all n E N. ylgn ihi

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Q: n the throwing of two di he same time, calculate dds of getting at least

A: The formula to compute odds ratio is,  Odds = p / (1-p) Here p is the probability of getting at…

Q: 9 checik whether the following statemects are true or False. lif En an is conditionally convergent,…

A: For (i), Let us take an=(-1)nn. So, the series ∑n=1∞an is conditionally convergent but ∑n=1∞an is…

Q: 7. Construct the generating functions for the number of partitions of n such that: a. No part is a…

A: Determine the generating functions for each part. Write the generating function of part (b) and then…

Q: 8. Let fi, f2, f3, ...,f, be Fibonacci numbers. Prove by mathematical induction that f = f,fn+1, i=1…

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Q: Use generating functions to find the number of ways to select 10 balls from a large pile of red,…

A: (a) Now since here we do not want less than two balls the coefficient for x0 and x1 for the…

Q: (). 2 4 3 The special sequence x is (select the most correct answer) None of these. Decreasing and…

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Q: 1 is divergent. all n, then n=1

A: Given ∫0∞f(x)dx diverges∑n=1∞an convergent

Q: 17. Find a combinatorial proof that n("z ¹) = (k+1)(²1). k 19

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Q: Use generating functions to find the number of ways to distribute r jelly beans among eight children…

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Q: Use generating functions to find the number of ways to select 50 balls from a large pile of white,…

A: The posted question has multiple questions bit according to guidelines we will solve the first…

Q: Sequence is defined by 02 U。く2 n70: Un = V2- Un-1 Suppose: Po-Von and an= = Uzn+1 * Prove that the…

A: Given: Sequence is defined by 0&lt;U0&lt;2 n&gt;0: Un=2-Un-1 Suppose : pn=U2n and  qn=U2n+1

Q: Let fn be the number of strings of length n using the letters A, B, C, D, such that there are an…

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Q: 2. Calculate Fn, when n=0, using the Binet form

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Q: (4) Determine the generating function for the sequence ho, h1, ..., where hn = #((a1, a2, a3, a4) |…

A: By definition of Generating Function,                The Generating function for the sequence…

Q: 3. Let's flip a fair coin independently an infinite number of times. Show with (one of the two…

A: Given the experiment that flipping a fair coin independently an infinite number of times.

Q: . (a) Show that if a, is a series of nonnegative terms that diverges to o, then any

A: Answer :-

Q: Use mathematical induction to prove the summation formula: 1 + 3 + 5 + . . . + (2n − 1) = n2

A: The solution is given as

Q: Exercise 6. Let (xn) be a sequence with x1 1 and xn+1 V6 + xn for n > 1. Prove xn 3.

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.