The common difference of an arithmetic sequence is the difference of two consecutive terms, . It can be calculated from the difference of any two terms  and  from the formula

By Statement 1 alone, setting and ,

.

By Statement 2 alone, setting  and ,

.

, so the problem is equivalent to solving 

.

It can be seen that ; since the expressions within the two sets of absolute value bars are opposites, it follows that

, 

and the equation is equivalent to

This is the only solution, so the correct choice is one.

, so the problem is equivalent to solving 

.

It can be seen that ; since the expressions within the two sets of absolute value bars are opposites, it follows that

, 

and the equation is equivalent to

Therefore, the statement  is true for all real values of , and the correct response is infinitely many.

A function  has an inverse if and only if, if  are in the domain of , then   implies that , or, contrapositively, if , then . In each of the four choices, no two values of  are matched with the same value of , so  implies that  in all four choices. Since the entire domain is given to be , all of the given functions have inverses.

A relation is a function if and only if, for each value  in the domain, there is one and only one value in the range that can be paired with . Let us examine all of the relations.

Pairing each positive integer  with the word "orange" if  is odd, and the word "turquoise" if  is is divisible by 2:

Every positive integer is either odd (not divisible by 2) or even (divisible by 2); no integer is both. 

Pairing each positive integer  with the word "brown" if  and 20 are relatively prime, and the word "gray" if  divisible by either 2, 5, or both:

If  is relatively prime to 20, then it shares only one factor with 20, which is 1; otherwise,  shares at least one prime factor with 20 - 2, 5, or both.

Pairing each positive integer  with the word "yellow" if  has four or more factors, and the word "black" if  has fewer than four factors:

This is simply saying that either  does fall in a set or does not fall in the same set.

Pairing each positive integer  with with the word "green" if  is a prime integer, the word "red" if  is a composite integer, and the word "white" if .

Every positive integer except 1 is either prime or composite, and no number is both; 1 is considered neither. 

In each of the above cases, each positive integer  belongs in exactly one of the described sets, so  is paired with only one word. Each of these relations is a function. 

Now consider this relation:

Pairing each positive integer  with the word "blue" if there exists a positive number with absolute value , and the word "purple" if there exists a negative number with absolute value .

For each positive integer , there exists one positive integer and one negative integer with absolute value ; for example, if , then 

.

Therefore, each positive integer  is paired with two values, "blue" and "purple", and the relation is not a function.

Substitute  and  for  and  in the expression for :