Obtain the generating function for the number of k-part compositions of n in which the ith part does not exceed 5i for all i = 1, 2, ... , k.

Q: 1. Let p| (a1a2 · .. an), then 3i with 1<i<n such that p | a;. be a prime. Prove by induction on n…

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Q: 10. Prove that if o is the m – cycle (a,a2 ... am), then for all i e {1,2, ...m}, o'(ar) = ar+i,…

A: Given that σ is the m- cycle (a1,a2,...am). We need to prove (i) σi(ak)=ak+i mod m for all i ∈{1, 2,…

Q: 2. A derangement is a permutation TE Sn with no fixed points, i.e. none of the elements remain in…

A: Introduction: Derangement is a very significant in combinatorics. Derangement is the process where…

Q: Show that for all n> 14, n cents can be represented by some combination of 3-cent coins and 8-cent…

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Q: Let S be the subset of (0, 1) consisting of numbers whose digits (in the decimal expansion)…

A: I have just used a property of countable set

Q: Prove by induction that for all n > 2, Ln = Fn-1 + Fn+1 Where La denotes the nth Lucas number and Fn…

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Q: Given any whole number w, how many whole number triple decompositions does w have? Prove it.

A: A number is said to be whole when it begins from 0 and has no end. Decomposition of number means…

Q: Make an outline for a proof by induction that 2 = n(n+1)(2n + 1) for all n E Z+. k=1

A: Note: We can solve one question at a time so kindly upload second question as another question.

Q: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: Let p be a prime. Show that there is exactly two elements a belongs to Zp s.t a^2 = 1.

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Q: 15n + 2 Let {an} be a sequence such that sn = %3D Зп — 9 - Compute > an · n=1 If this is divergent,…

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Q: 4. Let A = of O 2| Calculate

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Q: Find an integer n such that U(n) is isomorphic to Z2 ⊕ Z4 ⊕ Z9

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Q: (a) Let m and n be integers greater than 1. What is the order of the element (1,1) in Z/mZ x Z/nZ?…

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Q: Find the order of the residue class 5 in Z/29Z.

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Q: Prove: For n > 1, n − 1 is a generator of Zn.

A: Introduction: A cyclic group or monogenous group is a group that is generated by a single element,…

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

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Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: Find the generating function for the following recursion relation with initial conditions ao=0 and…

A: Let, the generating function be  fx=a0+a1x+a2x2+a3x3+a4x4+a5x5+a6x6+... Given,  a0=0a1=3…

Q: 4. The generating function f(x) for the Fibonacci numbers 1, 1,2,3,5,8, 13,21, ... is given by OA.…

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Q: Show that the last digit of the Fermat number Fn =2^(2^n)  is 7 for all integers n ≥ 2. [Hint: By…

A: We have to prove that the last digit of the Fermat number Fn=22n is 7 for all integers n≥2. For…

Q: 4. Let a 1. 2 Follow these steps to show that (x„)n€N is Cauchy. (a) Draw a picture and let L = b –…

A: Given - Let a &lt; b. Let x1 = a, x2 = b, and xn+2 = xn+1 + xn2 for n ≥ 1. To Show - Use induction…

Q: Derive a formula for E-1 Qiwhere Qi represents number of edges of n-dimensional cube and prove the…

A: As indicated, we shall use mathematical induction.

Q: Prove the following formula for all n E N by induction: 1+ 2·2 + ... +n· 2"-l = 1 + (n – 1)2".

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Q: Suppose that phi (x) belongs to Aut (Zn) and a is relatively prime to n.  If phi (a)=b, determine a…

A: According to the given information it is given that:

Q: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

Q: Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n Σ'=1+2+...+ B₁=k²= 1² +2²+. k=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: b) Suppose n> 1, and m is relatively prime to n. Show that m is a generator of Zn. c) Suppose n> 1,…

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Q: A shop gives gift cards in nominal 25 and 40k. determine the possible total amount that can be…

A: Proof at first, the amounts seem semi-random:

Q: Exercise 15.5.10. (a) Show that Z, is cyclic for any integer n > 1 by identifying a number a such…

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Q: Consider the following theorem. Theorem 9.5.1: The number of subsets of size r that can be chosen…

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Q: 2n 2n 1/2n + 2 1, 2 n +1 n -

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Q: 5. Find the generating function to determine the number of ways to pick k objects from 2n objects…

A: Given: When duplicates are allowed and the (2i – 1)th and (2i)th objects occur the same number of…

Q: Prove that Fisher's ideal index number lies between Laspeyre's and Paasche's index number

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Q: 7. Prove the formula for alln EN using induction. n j(j+1) n +1 j=1

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Q: Find a generating function for the sequence (ao, a1,...) where a, is the number of nonnegative…

A: To find- Find a generating function for the sequence a0, a1, ... where ar is the number of…

Q: Let n E Z. Prove that n3 – 7n + 18 is divisible by 6 for every interger n. Do not use induction.

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Q: (a) Determine whether 2 is a primitive element in Z37. (b) Find an element of Z37 that has order 12.

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Q: Give a combinatorial proof: Forn and k satisfying 1 <k < n, n nk = Ek · (j – 1)k-1. j=k

A: The complete solutions are hoben below

Q: Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an…

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Q: Let 5 be a primitive 9th root of unity in C and consider the extension Q(5) of Q. Prove that T; is a…

A: The Galois extension field is very significant in field theory. For a primitive nth root of unity,…

Q: Q/ Z is infinite, but every element of Q/ Z has a finite order.

A: ℚℤ is infinite, but every element of ℚℤhas a finite order.Now consider the group of rational number…

Q: Use mathematical induction to prove that for z>0 and any positive integer n, e21+z+ 2! n!

A: Recall that,    If f'(x)≤0 in [a,b] then f(b)≤f(a)  and   If f'(x)≥0 in [a,b] then  f(b)≥f(a)

Q: . Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, ø(m)) = 1, %3D where…

A: Here GCD(b,m)=1 means they are coprime.

Q: (This exercise constructs another proof of the infinitude of pimes) Show that the integer Qn = n! +…

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Q: Find a generating function for a,, the number of partitions of r into (a) Even integers (b) Distinct…

A: a. The polynomial ( 1 + x^2 + x^4 + ...) represents the number of 2's in our partition. The…

Q: 3. For every integer k > 0, define A = {2*n : neN and n is odd}. Prove that {Ag : k = 0, 1, 2, 3,…

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