8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D. a) Find the first two integers K and D that satisfy these conditions and then b) Evaluate the expression A in both cases. A= [1+2+ .. + (VK – 1)] – [(VK +1) + (VK +2) +... + D] c) Can you make a conjecture about the value of A? Does it depend on the values of K and D?

Q: . Rewrite the equality of ideals (r+ y/2) = (3 – 4/2) in Z[/2] in terms of elements.

A: We know that, A subring  B of a ring R is called a ideal of R if for every r∈R and for every b∈B,…

Q: 5**) (a) Prove that for all positive real numbers n, r, and s, if rs < n, then r < Vn or s < Vn.

A: As per Bartleby's answering policy, we can answer only one question, so kindly post the next…

Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

A:

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A: We have to prove A is contain in B and vise versa

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A: Solve the question by first opening the summation on the left side of the inequality. Simplify the…

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A: We have to prove given identity:

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A: Note that there is a typo in the question. We need to show that,                            We will…

Q: (b) Let A¡ = [-i, 1] for every positive integer i. Find U A¡ and N1 Aj.

A: Given that Ai=-i,1 for every positive integer i. That is, A1=-1,1, A2=-2,1, A3=-3,1 and so on. To…

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

A:

Q: a) Prove that, for all integers K > 5 Р(к +1,5) — Р(К,5) %3D 5P (К, 4)

A:

Q: Show that a fraction r = a/ b in lowest terms has a finite decimalexpansion if and only ifb = 2n5m…

A:

Q: a. Prove that the v63 iş irrational b. For all integers, prove that 4 (n2 + 2) c. Prove or disprove:…

A: Note: As per our company guidelines we are supposed to answer the first three sub-parts of the…

Q: -), prove that for all n E N: (a) n-1 E(:)-E- n k k=1 k=0 (b) ¿(:)-- = 2" k k=0 (c) El-1)* E-1» (:)…

A: According to Bartleby policy we solve up to 3-subparts. Please repost #d and mention it.   b)…

Q: Disprove the statement by giving a counterexample: For all integers m and n, if 2m + n is odd then m…

A:

Q: Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x2 +…

A: We already know, (x,y,z) = (3,4,5) is a solution of the equation, x² + y² = z² i.e. 3²+4² = 5² Now,…

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

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A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

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: 1. Find an r> 0 such that all zeroes of 24 + 6z²+z+3 are contained in D,(0).

A:

Q: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

A:

Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

A:

Q: Prove T(n)= 2T(n-1)+n can be in 0(n2")using represented telescoping method

A:

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: 5. Prove that for all n n [j³ = j=1 n² (n + 1)² 4

A:

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: 2. Show that if x and y are perfect squares (that is, x = m? and y = n2, where m and n are…

A:

Q: 2. ) Show that the cube of any integer is of the form 7k or 7k +1 b) Verify that if an integer is…

A: 2) a. To show that ''The cube of any integer is of the form 7k or 7k±1''

Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

Q: 28. Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(Vd) where d is not…

A: Given [E:D]=2.

Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

A:

Q: if n is a perfect square then n2 + 2n is not a perfect square. prove this

A:

Q: 4. Let R be the set of all real numbers, xER and n EN such that n > 2. If (1+x) >0, prove that Pn :…

A: Mathematical induction.

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: If nis a positive integer 1 then prove (D - a) n!

A:

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

A:

Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: prove that and = nd(n)/2

A:

Q: tn be a positive integer and M = [2 01 %3D en M" 2n equal to the above None of the mentioned

A:

Q: (4) Prove that if s and t are odd perfect squares, s + t is not a perfect square. Use contradiction…

A: A square of a number is the product of a number with the number itself . A perfect square is the…

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

A:

Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

A:

Q: 2 Lk=0 4. Prove that for all positive integers n, п

A:

Q: Prove that for any positive integer n, Vn is either an integer or irrational.

A:

Q: 2) Let t be the nth triangular number. Prove that t +3 n + 6 is n triangular.

A:

Q: [2.1.22] I have 51 rectangular pieces of cardboard, each of which has an integer length and width in…

A: Given: Let's consider I have 51 rectangular pieces of cardboard in set 1,...,100, each with an…

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…