Show that the permutation [n, n-1, ... , 2, 1] has n(n-1) inversions.

Q: Solve the recurrence relation an+2 - 5an+1 + 6a, = 2, n = 0, 1, 2, ... , a, = 3, a1 = 7 using…

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Q: (b) Use mathematical induction method to show that 품(주-)- (두-0) (주-)-(-)1 n+1 2n

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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…

A: We can prove this by induction method

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: 1. A restaurant is preparing a large round table with 2 +1 seats (the seats are unla- belled). The…

A: Given, Number of available seats = 2n+1 Number of red napkins = n+1  Number of blue napkins = n

Q: The distinct characteristic roots corresponding to the recurrence relation an=7an-1 -10an-2 are:…

A: The given recurrence relation is an=7an-1-10an-2. Rewrite the recurrence relation is an=7an-1-10an-2…

Q: Theorem. If n is composite, then there exists at least one pair of zero divisors in Zn.

A: This is quite a simple question and can be solved using basic definition of zero divisor and prime…

Q: Prove by the method of induction for n >1: 1, 1 1 1 +- 1-3 3.5 5-7 (2n-1)(2n+1) 2n+1

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Q: Let r(n) be the number of n-permutations whose square is the identity permutation. Prove that if n >…

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Q: Prove that ?? is generated by the following n-1 transpositions: (1 2), (2 3), (3 4),. . . , (n-1 n)

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Q: Example 3 A play has a cast of 7 actors preparing to make their eurtain call. Use the permutation…

A: A play has cast of 7 actors preparing to make their curtain call.Use the permutation formula to find…

Q: 2 3 4 5 1 3 5 (1 2 3 4 5 Consider the permutation 12 4 2. 3 4 1 1. Express as a product of disjoint…

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Q: Use mathematical induction to prove that. 1+3+5...+(2n-1)=n2 for all n ≥1.

A: To prove using mathematical induction: 1+3+5…+2n-1=n2 for all n≥1. Proof: Let the given statement be…

Q: Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n = An Σ=1+2+. .+ n₂ =k²= 1² +2²+.…

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Q: Let o be a fixed odd permutation in Sn. Show that every odd permutation in S, can be written as a…

A: The main objective is to prove that every odd permutation can be expressed as σα for some α even…

Q: (a) Prove that the product of two even permutations is even. (b) Prove that the product of two odd…

A: ANSWER (b) Product of two odd permutations is even: Let us take two odd permutations i.e. f and g.…

Q: Express the permutation (2 4 5)(1 3 5 5)(1 25) as a single cycle or as a product of cycles.

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

Q: Prove the following by using the principle of mathematical induction for all n ∈ N: x2n– y2n is…

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Q: (a) Express the permutation (2 4 5) (1 3 5 5)(1 25) as a single cycle. or as a product of cycles.

A: Note :-  As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: Use mathematical induction to prove that 1 1 1 n 1.2 2.3 3.4 п(п + 1) n + 1 for all positive…

A: To prove: The sequence 11·2+12·3+13·4+...+1nn+1=nn+1 by mathematical induction for all positive…

Q: Exercise 14.6.11. Suppose a is an an even or odd permutation? n-cycle. How can you tell whether a is

A: EX 14.6.11 : If σ=(a1 a2 a3....an), We need to explain how can we identify whether σ is an even or…

Q: # 2. Use mathematical induction II to prove that for all integers n 2 5, 1 + 4n < 2".

A: Given: n≥5 To prove: 1+4n&lt;2n using mathematical induction.

Q: 6. Prove that the product of an even permutation and an odd permutation is odd.

A: Let S be set on which permutations are defined. Let  α=(x1,x2,...,xk) and  σ=(y1,y2,...yn) be two…

Q: Let aj = 1 and 1 -ап, п >1. n+1 аn+1 Show by mathematical induction that for all positive integers…

A: we have to solve by using mathematical induction

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

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Q: 3. Suppose a, nEZ and 0 < a< n. Prove that - u 1 + a a (:) in terms of factorials; and secondly by…

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Q: Prove using mathematical induction that for all n ≥ 1, n(3n-1) 1+4+7++ (3n - 2) = 2

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Q: Find the number of inversions in the permutation 52134 of Sz = {1,2, 3, 4, 5}. 6 O 5

A: Number of inversions

Q: 5) Let a, = 2a +5,a, =1, use induction to show that a = 6(2")-5 for all n20.

A: Given that an=2an-1+5 and a0=1. We have to show that an=62n-5 for all n≥0. We have to use the…

Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: aly (3) 2.3-1=3" – 1 i=1

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Q: 2. Let Qn be the number of permutations of {1,2, .., n} in which none of the patterns 12, 23, .., (n…

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Q: Use mathematical induction to prove that for all integers n > 5, 1+4n < 2".

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Q: Let o be a fixed odd permutation in Sn. Show that every odd permutation in Sn can be written as a…

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Q: (1) Find the recurrence relation of an = 6an-1 - 9an-2 with initial conditions ao = 1 and a1 = 6?

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Q: Prove or disprove: For n > 2, n − 1 is a generator of U(n)

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Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: Consider all n! permutations of the set {1, 2, 3, . . . , n}. We say that a permutation has i fixed…

A: a) n=4   Permutation on the set {1,2,3,4} number of permutation have 4 fixed points can be found by…

Q: 8. (Chapter 10, #16) Prove that for every positive integer n, 2" +1< 3".

A: Using induction to prove any statement includes first proving the statement is true for n=1, then…

Q: Prove using mathematical induction that for all n > 1, n(3n – 1) 1+4+7+·..+ (3n – 2) 2

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Q: Use mathmatical induction to prove that for all intergers n is greater than or equal to 1.…

A: For n = 1,       LHS = 4 [only 1 term]      RHS = 2.12 + 2.1 = 2.1 + 2.1 = 4 Hence, LHS = RHS.  So,…

Q: Mathe matical Induchion Prove that n°-n is diviside lag 6. 3 n -n

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Q: 1. Use induction to prove that Σ m(n+1) for all n eN. k=0

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Q: prove using mathematical induction that for all n ≥ 1 1+5+9+13....+(4n-3)=2n² -n

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Q: Consider all n! permutations of the set {1,2,3,..., n}. We say that a number i is a fixed point of a…

A: Given: pn(k) be number of permutations of 1, 2, ..., n that has exactly k fixed points.

Q: 7. Prove that a permutation in S. n ≥ 2, cannot be both even and odd.

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Q: Use mathematical induction to prove that for all integers n ≥ 1, 1 +7 +13 + 19 + ... + (6n − 5) =…

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Q: Use mathematical induction to prove that:   3n ≥ 2n + 1 for n ≥ 1

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Q: I. Consider the permutation (1 2 3)(4 5)(1 25 4)-2. 1. Express as a product of disjoint cycles. 2.…

A: consider  the permutation (123)(45)(1254)-2

Q: Prove by induction that for all n ∈ N with n ≥ 8, 2^n ≥ 3n^2 + 5

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