5. Prove that (n, n + 2) = 1 is n is odd and (n, n + 2) = 2 is n is even.

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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

A: .

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

Q: Show that 1+2+3+...+n= ("""). (";"').

A:

Q: 7. Prove that if a, = 1, a1 = 4, and a, = 6a,-1 – 9an-2 for n > 2, then a, = 3"-(3 + n) for all n 2…

A: We have to prove the following result by using the principle of mathematical induction.

Q: 7 Disprove the following statements: (a) If n and k are positive integers, then n-n is always…

A: Given Data: To disprove the following statements.

Q: 3 21. (k +1) k=1

A: The given expression is, ∑k=13(k2+1) To find the sum of the expression from k=1 to k=3 substitute…

Q: 4. Conclude that n2 n=1 ||

A:

Q: 5. Prove that, for n> 1, n

A:

Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

A:

Q: 81. Prove: 2+4 +6+ + 2n = n(n + 1). ...

A:

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: . Prove that 3+3.5+3.5²++3.5"= whenever n is a nonnegative integer.

A: Sum of a n terms of GP is  where 'a' is the first term and 'r' is the common ratio. Where 'n' is non…

Q: Below is an incomplete proof Prove that 3n2 > 2(n+1)2 for n ≥ 5 3n2 > 2(n+1)2 Is equivalent to 3n2 >…

A: The question already shows how the proof for 3n2 > 2(n+1)2 for n≥5 is equivalent to showing n2…

Q: B. Find the value of n: (n+3)t 1. (n+1)! = 20

A:

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: 16. What is the greatest integer N such that (10 – N) + (16 - N)> 5?

A:

Q: In the Powerball Lottery, 5 different numbers from 1 to 69 are selected. The order they’re selected…

A: There are  different numbers from  to  that are selected in the Powerball lottery having the order…

Q: 2. Find n and a if: Inx² + In2= In ax"+1

A: Given equation is lnx2+ln2=lnaxn+1. We have to find the value of n and a. Used formula: lnmn=nlnm…

Q: What is the converse of the statement "If n is an even integer, then 5n - 3 is an odd integer."?

A: A statement which is  obtained by reversing the given hypothesis and conclusion of a conditional…

Q: 10n+1 + 10" +1

A:

Q: The access code for a car security system consists of four digits. Each digit can be 0 to 9, How…

A: Givena car security system consists of 4 digits each digit can be 0 to 9

Q: 1. For any integer n, the quantity 3 (n² + 2n + 3) - 2n² is a perfect square.

A: We need to show that for any integer n following quantity is a perfect square: 3(n2+2n+3)-2n2

Q: 6. For any positive integer n > 1 one has: (2n)! (n!)² n+1 <

A:

Q: V5n – 10 = n- 2

A: Square both sides and solve the quadratic equation to get the value of n.

Q: Solve for n in permutation 9(n²-1Ps) + 9(7²P6) = n²+1 P7 %3D

A: Let's find.

Q: For how many integers n with 0 ≤ n ≤ 100 is the quantity integer? n(n-1) (2n-1) 12 equal to an

A:

Q: If x – 1 is an even integer, which of the following numbers must be odd? I. x(x - 1) II. 3(x - 1)…

A:

Q: 10. Prove that n³-n is divisible by 6 for every integer n.

A:

Q: 5. Prove that (n,n+2) = 1 is n is odd and (n, n + 2) = 2 is n is even.

A:

Q: It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2)…

A: