Let n EU = Z and p(n): n² + n is an even integer. a. Write the following statement: (Vn) p(n). b. Is it true or false? c. Prove your answer in (b).

Q: For any positive integer n, n(n+1)(2n+ 1) EP = 12 +22 + .. + r?

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Q: n! %3D if n is non-negative integer and proof r-1 r>t

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

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Q: Let A = {n ∈ Z; n is odd} and B = {n ∈ Z; 3n 2 + 7 is even}. Prove that A = B.

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

Q: Define P(n) to be the assertion that: Σ n(n + 1)(2n + 1) j=1 (a) Verify that P(3) is true. (b)…

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Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: Did I prove p is not equal to 0 correctly? p is not equal to 0 by what?

A: You have proved correctly .   This can also be seen as If p= 0  Then 1/x = 0  and x= infinite or…

Q: Analyze the following "proof" and determine if there is anything incorrect and why. Claim. Suppose m…

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Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

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Q: Prove that for any nonnegative integer n. n + +.. + 22m i+1 i +2 n+1 i=0 the

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Q: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: Q5 Show that EL >². n n =1

A: Consider that x is a random variable. From the formula of population variance;

Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

Q: d x" = nx"-1 de 4. Prove that NɛQ

A: ddxxn=nxn-1 ;   n∈Q

Q: for n>2, 5 . Deermine for whpch n, n2-2 let |anlく,

A: Given  an = 5nn2-2

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

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Q: What is the general rule to evaluate I N,where N is any positive

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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: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

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Q: (b) Given integers a, b and c such that cla and cb then c|(ar + bs) for all r, s E Z. True False &…

A: Given that c|a and c|b for some given integers a,b and c The statement is c|ar+bs for all r,s∈ℤ…

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: a) Suppose, T(n) = 1²+2²+3²+..+n2. Show that, T(n) is in O(n³). %3D .....

A: Big-⊖Notation: Let f and g be real valued functions and assume g is eventually position. We say that…

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

Q: 14b. Prove by mathematical induction: Vn E N, n³ –n is a multiple of 3. -

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Q: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

Q: Prove the statement for positive integer n and r, with r ≤ n. P(n, 1) = n

A: To prove the statement

Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: II. Prove the following using Direct Proof. 6. If x is an odd number and y is even number, then x +…

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Q: 9.59. a. Let (a][b] € Zs. If [a] · [b] = [0], does it follow that [a] = [0] or (b] = [0]? b. How is…

A: Prime number p has factor 1 and p

Q: 2) If n is an odd integer, then H= (n-1) 2

A: To prove that n2=n-12 such that n is an odd integer.

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

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Q: Q2/ prove [A] [A] = [I] if [2 4 -5 [A]= 2 7 1 3 4

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Q: | +スt3+ *n = nintl) for all 1 てU

A: The given formula is                                    1+2+3+..........+n=n(n+1)2       for all n≥1…

Q: Let X~N(3,0²). Suppose that P(3<X<6)=0.4 · find P(X<0) ?

A: Given: X~N3, σ2 and P3&lt;X&lt;6=0.4.

Q: Define P(n) to be the assertion that: n(n + 1)(2n + 1) 6 j= (a) Verify that P(3) is true. (b)…

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Q: For a nonnegative integer n, prove the following equation using mathematical induction. (? +) (n +…

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Q: Q2(a) For which positive integers n is 3n + 4 < 2"? Prove the inequality using mathematical…

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Q: 5. For non-negative integer n and r, where 1<rsn, show that (n-r+1)"C,1=n(""C,). r-1

A: Given n and r non negative integers with  1≤ r≤ n. We need to prove (n-r+1)nCr-1=n(n-1Cr-1)

Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: - Let n and r be positive integers such that r≤n. Then (a) P = nx pn-1 (b) P = P-1

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Q: 8. Prove 3n e R|n? – n < 0

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Q: Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.

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Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is  ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

Q: prove n2 + 5 greater than or equal to 4n for all integers n

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Q: (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < a +b < b. (b) Prove by contradiction : For…

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Q: 9.59. a. Let (a][b] E Zs. If [a] · [b] = [0], does it follow that [a] = [0] or [b] = [0]? b. How is…

A: Solution by using definition of zero divisors:

Q: (3b) Let n E Z. Prove that if 3|(2n² + 1), then 3 ł n

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Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…