Discrete Mathematics Answers

Prove that the product of any three consecutive integers is a multiple of 3. 

Use a proof by contraposition to show that if 𝑛𝑛2 + 1 is even, then 𝑛𝑛 is odd.

Use a direct proof to show that every odd integer is the difference of two squares.  

2. Use generating functions to solve the recurrence relation a_n = 4a_n−1 − 4a_n−2 + n², where a₀ = 2, a₁ = 5.

List the quadruples in the relation { a,b,c,d} where a, b, c, d are integers with

0 < a < b < c < d < 8.

Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0,1),(1, 1), (1, 2), (2, 0), (2, 2) and (3, 0). Find the

(i) reflexive closure of R, (ii) symmetric closure of R

Draw the directed graph that represents the relation {(a,a), (a, b), (b, c), (c, d), (a, d), (b, d), (d, b)}.

Suppose a recurrence relation

an=2an−1−an−2

where a1=7 and a2=10

can be represented in explicit formula, either as:

Formula 1:

an=pxn+qnxn

       or  

Formula 2:

an=pxn+qyn

where 

x

and

y

are roots of the characteristic equation.

  Determine p and q

List the quadruples in the relation { a,b,c,d} where a, b, c, d are integers with

0 < a < b < c < d < 8.

Determine whether the following relation is reflexive, symmetric, antisymmetric and/or transitive.

(x, y) ∈ R if x ≥ y, where R is the set of positive integers.