n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: Find the value of c if -226 Σ1+0 2 C = No decimal entries allowed.

A: summation (1+c)^-n from 2 to infinity converges only if |1+c|<1(1+c)^-n is a GP series. Use the…

Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: e havel (² + ¹) - (2) - (2) = Prove that for all positive integers x and y we xy

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Q: Consider the sum (3)² + (8)² + (13)2 + · . + (15n – 2)2. | m 1 1 m(т + 1)] and k2 = (m(m + 1)(2m +…

A: Consider S=32+82+132+ . . . .+15n-22

Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

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Q: 3. Prove or disprove. For n> 1, use mathematical induction to establish each of the following…

A: Let's say we need to prove a formula or a statement P(n). To prove this, we need to prove that The…

Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

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Q: Let r be a real number and let k be an integer. We then define the binomial coefficient () by r(r-1)…

A: Given, r is a real number and k is an integer  The Binomial Coefficient rk by rk= r (r - 1) --- (r -…

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: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

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Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

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Q: The numerical value of xe dr is

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: Show that when |z +1| < 1,z² =1+£(n +1)(z+1)" n=1

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Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: 7

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Q: 3. Prove each of these divisibility statements by mathematical induction. (b) (a) 8|52n +7 for all n…

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Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Show that if s(3) 4(q- _9−2+9(1+ 93 2(g-1) increasing or decreasing. For 3 > 0, q> 3

A: The given function is                               sβ=q-2+q1+4q-1qβ122q-1 where β&gt;0 and q&gt;3.…

Q: (a) There exists two odd integers whose average is odd. 1 1 (b) There exists distinct positive…

A: (a) ans:- 1,5 (b) ans:- 2,3 &amp;6 For detail...go through the solution

Q: use gauss's trick to determine R(n)=2+3+4+...+(n-1)+n+(n+1)

A: It is known that sum of first n numbers is n(n+1)2  that is  1+2+3+----+n=n(n+1)2 Gauss trick method…

Q: Find the generalized union and intersection: 3) A = {1,2,3,...} A₂ = {x|x = 22m where m eset of…

A: Given that, Λ={1,2,3,…} Λλ=x|x=2λm where m∈set of integers from -λ to +λ The objective is to find…

Q: ove or disprove: There exists an integer a such that 14 (3a-5) and 21 (2a +

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Q: Prove that 1¹x 2² x 3³ x...xn" < n(n+1) 2 (2n+1) 3

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Q: Prove for every positive integer n that 2n 2n (²n) = 2(²-1).

A: We have to how that for every positive integer n

Q: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers equals…

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Q: Which of the following statement is always false? (a) For all integer x, [x] = [-x] (b) For all odd…

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Q: Prove by mathematical induction that for every positive integer n, 1 1 1 1 П 3i 3і Зі + 1 (Зп + 1)!…

A: Since you have posted multiple questions and we can answer one question at a time. So, answered…

Q: Prove that (n+1)C n is equal to n+1

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Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…