Question

to make a number pattern, yuan used the rule \add 5,\ with a whole number as the starting point. which statement must be true about yuan’s list of numbers? each number is a multiple of 5. the numbers alternate between even and odd. it has an even number that is prime. none of these statements must be true.

Explanation:

Step1: Analyze the rule "Add 5"

If we start with a whole number \( n \), the sequence will be \( n, n + 5, n + 10, n + 15, \dots \)

Step2: Check the first statement "Each number is a multiple of 5"

Let's take an example. If the starting number \( n = 1 \) (a whole number), the sequence is \( 1, 6, 11, 16, \dots \). None of these are multiples of 5. So this statement is not always true.

Step3: Check the second statement "The numbers alternate between even and odd"

Case 1: If the starting number \( n \) is even (e.g., \( n = 2 \)), the sequence is \( 2, 7, 12, 17, \dots \). 2 (even), 7 (odd), 12 (even), 17 (odd) - alternates.
Case 2: If the starting number \( n \) is odd (e.g., \( n = 3 \)), the sequence is \( 3, 8, 13, 18, \dots \). 3 (odd), 8 (even), 13 (odd), 18 (even) - alternates.
Since adding 5 (an odd number) to an even number gives an odd number (\( \text{even} + \text{odd}=\text{odd} \)) and adding 5 to an odd number gives an even number (\( \text{odd} + \text{odd}=\text{even} \)), the numbers will always alternate between even and odd.

Step4: Check the third statement "It has an even number that is prime"

The only even prime number is 2. If the starting number is greater than 2, say \( n = 3 \), the sequence is \( 3, 8, 13, 18, \dots \). 8 and 18 are even but not prime. If \( n = 2 \), the sequence is \( 2, 7, 12, 17, \dots \). Here 2 is prime, but if \( n = 4 \), the sequence is \( 4, 9, 14, 19, \dots \). 4 and 14 are even but not prime. So this statement is not always true.

Step5: Check the fourth statement "None of these statements must be true"

Since we saw that the second statement "The numbers alternate between even and odd" must be true (as shown in Step 3), this statement is false.

Answer:

The statement "The numbers alternate between even and odd" must be true. (Assuming the options are labeled as: Top - left: "The numbers alternate between even and odd", Top - right: "Each number is a multiple of 5", Bottom - right: "It has an even number that is prime", Bottom - left: "None of these statements must be true", so the answer is the top - left option: "The numbers alternate between even and odd")