1. If t, and s, are trinagular and square numbers, respectively, show that 8t, +1= s2n+1- 2. A pentagonal number p, is defined by n(3n – 1) Pn a. Determine and illustrate the first four pentagonal numbers. b. Define pn recursively.

Q: 2. Let N = {A, | ne N}, where A, = {n,n+ 1, n +2, ...}. (a) Find UN and prove your conjecture. (b)…

A: we have given Ω = An n∈N where An =n, n+1, n+2,...... we have to find (a) find ∪Ω and prove your…

Q: .. with n, m, E No for all i e {1,...,r}. Let a = p p... p and b p P (The existence of such a…

A: Let a=p1n1.p2n2.....prnr and b=p1m1.p2m2.....prmr with ni,mi∈ℕ0 For all i∈{1,.....,r}. To prove…

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: Define an by an for 4an-1 +5an-2, n 2, ao = 5, a1 = 13. Find an explicit formula %3D an

A: an=4an-1+5an-2 an-4an-1-5an-2=0 for the homogeneous system, write the auxiliary equation  r2-4r-5=0…

Q: see the question as attacheed here

A: Solution: If we considers the first several values of Tn-n! it looks like Tn=n!+2n, which we now…

Q: b. Suppose that {a, }is a sequence of real numbers defined recursively by Untl Vn z1 Write out the…

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Q: Find an explicit formula for an if ao = 1 and an (n + 1)an-1+3" %3D for n > 1.

A: To Determine :-  An explicit formula for an if a0 = 1 and an =  n + 1an-1 + 3n for n ≥ 1.

Q: Show Hhat the series Canverges to 1 Kei (K+)!

A: Introduction: The ratio test is one of the basic tests to verify the convergence of a given series.…

Q: 3. Find two elements a and b in Z25 such that a and b are units, but a +b is not a unit. Justify…

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Q: Show that 1+2+3+… +n = (";").

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

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Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: Question 1. Let a,b e Z (with at least one of a, b non-zero) and let d = gcd(a, b). Prove that gcd =…

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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 a closed formula for a, if ak = 5n² + 4n k=1 an =

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Q: Question 6. Let n, m, k be nonnegative integers such that m< k< n. Show that OC) -OG) k k – m

A: To prove: nkkm=nmn-mk-m We know that nk=n!k!n-k!

Q: 3. Let A={n €Z, 2Z). Prove or disprove: For every n E A we have n² = 4.

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Q: 2. Let x (1 3 4) and y= (1 24 3). (Note that these are the same permutations as in th previous…

A: (a) Here given x=1 3 4 =12343241 Thus  x-1=1 4 3=12344213 Because  1234324112344213=12341234=I

Q: 1. In this problem, you will prove the following Theorem: r E C if and only if there exists a…

A: (a.) To prove: x∈Ci if and only if xi∈0,2. (b.) To prove: x∈C if and only if there exists a ternary…

Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: 6.26. Let n > 2. Show that every element of S, can be written as a product of transpositions of the…

A: TO PROVE: Every element of Sn can be written as a product of transpositions. PROOF: By a theorem of…

Q: For m an n non-negative integers with 0 < m <n, give a combinatorial proof for the identity EOC) m…

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Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

A: Given:   If M and N are both bounded subsets of R. M+N :=m+n : m∈M  and  n∈N

Q: B. 20. Show that the subset R= {0, 3, 6, 9, 12, 15} of Zu is a subring. Does Rhave an identity? 21.…

A: Show that the subset S=0,2,4,6,8 of ℤ10 is a subring. Does S has an identity element?

Q: Let m eN and a, b e Z be such that (a, m) = 1 and (b, m) Let h = 0m(a), k = 0m(b), and let n = 0m…

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Q: Find an explicit formula for an if ao = 1 and an = (n+1)an-1+3" %3D for n 2 1.

A: Consider the provided question, If a0=1  and  an=n+1an-1+3n for  n≥1Now,…

Q: 1. If n is an integer, then 2n +1 is odd. 2. The square of any integer can be written in the form 3k…

A: 1) Given:     If n is an integer, then 2n+1 is odd.   Let n=1  then 2n=21              =2

Q: 4. Let bo = 1, b1 = 3, and for each integer k > 2, we have br = 4bk-1 – 4bg–2 %3D Prove that for all…

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Q: 7. Find a number t such that In(4t +3) = 5 (Give EXACT value)

A: Given that ln(4t+3)=5 we need to find "t"

Q: 3. For n e N, we define n! = (n)(n – 1)(n – 2)... (2)(1). E.g., 4! = (4)(3)(2)(1) = 24. Prove the…

A: Here we have to prove two statements.

Q: 27 en+1 – 1 < 0.05 ? 2 (n + 1) · (n + 1)!

A: Given: The expression is, 272en+1-1n+1n+1!. Calculation: Obtain the value n such that…

Q: 2.19 Transform the following parabolic PDES to canonical form: (a) + 9- ex = 1 ax? дх ду ду? ax a?u…

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Q: Find an explicit formula for an if a0= 1, a1= 2, and an= 2an−1−an−2for n ≥ 2.

A: given a0= 1, a1= 2  and an= 2an-1−an-2for n ≥ 2.

Q: 4. Let bo = 1, b = 3, and for each integer k > 2, we have br = 4bx–1 – 4bk–2 Prove that for all…

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Q: Let n>1. A hexagonal number h, is of the form h, = n(2n – 1). a. Determine the first 5 hexagonal…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

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 H = {(1), (12)(34), (13)(24), (14)(23)}.  Find the left cosets of H in A4.  How many left cosets…

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Q: Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a…

A: Given dk is number of derangement of order k. We know that for a set 1,2,…,k, the derangement dk are…

Q: Find an explicit formula for the nth term for an, given the following recursive relation а, 3 2а,-1…

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Q: Now, from the definition of k(m), given by equation (1.135), we find on setting n = 0, 1, 2, ...,…

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Q: 5 М — 11 Find formulas for the entries of M", wheren is a positive integer. M" =

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Q: Let the numbers eo, e1, e2, ... be defined inductively as follows: eo = 12, ei = 29 en = 5en-1 –…

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Q: Show that the n-th tetrahedral number O_n is given by S_n= n(2n^2+1)/3

A: A tetrahedral number corresponds to a 3-Dfigure in the shape ofa tetrahedron made of spherical…

Q: 1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the…

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Q: Let the numbers e0, e1, e2, . . . be defined inductively as follows: e0 = 12, e1 = 29 en = 5en−1…

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Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: Question 4 Use Mathematical Induction to show that 512|8 · 5" +6. 32n – 80n – 14 for all n E N.

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Q: Question 8: Using combinatorial argument, prove that (:) --C) - »C) * 311 +n* 3

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Q: Question 5. a) Prove that if a, b, c are positive integers, and c|a but cłb, then for all x E Z we…

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