Final answer:
The correct statement for four consecutive odd numbers a, b, c, and d is that the differences b - a and d - c are always equal. This is because when expressed in terms of a, b = a + 2 and d = a + 6 while c = a + 4, thus making both differences equal to 2.
Step-by-step explanation:
Given that a, b, c, and d are four consecutive odd numbers, we can represent them as:
- a = a
- b = a + 2
- c = a + 4
- d = a + 6
Now to find out which statement must be true:
- a. The sum a + b + c + d is 12 greater than a
Adding the numbers: a + (a + 2) + (a + 4) + (a + 6) = 4a + 12, which is 12 plus 3 times a, not just 12 greater than a. So, this statement is incorrect. - b. The sum a + b + c + d is 6 greater than a
This does not follow from the sum we calculated, so it is also incorrect. - c. The differences b – a and d – c are always equal
Since b = a + 2 and d = a + 6, c = a + 4, the difference b – a = 2 and d – c = 2. Hence, this statement is true. - d. The differences c – b and d – a are always equal
The difference c – b = (a + 4) – (a + 2) = 2, but the difference d – a = (a + 6) – a = 6, which are not equal, therefore this statement is incorrect.
Therefore, the correct statement concerning the consecutive odd numbers is c. The differences b – a and d – c are always equal.