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Properties of Consecutive Numbers

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What Are Consecutive Numbers?

Algebra problems often ask about properties of consecutive numbers. Consecutive means numbers like 1, 2, 3 or 9, 10, 11.

3, 6, 9 are not consecutive numbers, but are consecutive multiples of 3. Basically it means that the numbers are adjacent integers. A question may also ask about consecutive even numbers or consecutive odd numbers. These are numbers like 2, 4, 6, 8, 10 or 13, 15, 17. We take one odd number, then the very next odd number after, etc.

So, how do we represent these numbers algebraically? If the question calls for consecutive numbers, I can let one of these numbers be x. Then, the next consecutive numbers will be x + 1, x + 2, etc.

I f the question calls for consecutive even numbers, we would have to “make sure” that the number we chose was even. We can do this by letting the first number be 2x instead of x. What would the next consecutive even number be? Be careful. 2x + 1 is not even. So, our next numbers would be 2x + 2 + 2x + 4. Similarly, consecutive odd numbers would be of the form 2x + 1, 2x + 3, etc.

Examples:

The sum of two consecutive numbers is 13, what are the numbers? Let the first number be x and the second number be x + 1.

Then:

x + ( x + 1) = 13
2x + 1 = 13
2x = 12
x = 6

So, our numbers are 6 and 7.

What if we had chosen our consecutive numbers differently from the start? Would we get a different answer? Let’s see: Let the first number be x - 3, and the second number be x - 4.
These numbers are still consecutive numbers: one comes directly after the other.

(x - 3) + (x - 4) = 13
2x - 7 = 13
2x = 20
x = 10

Interesting!!

We have x equal to 10 here and we had x equal to 6 above!! What’s going on? Well, substitute 10 x = into the variables we chose and see what happens.

10 - 3 = 7
10 - 4 = 6
We get the same answer. Everything works out. Sometimes it makes things easier if you choose different variables for your consecutive numbers. For example, if you had a problem involving the product of 5 consecutive numbers, which would you rather calculate?

x (x + 1) (x + 2) (x + 3) (x + 4)

or

(x - 2) ( x - 1) (x) (x + 1) (x + 2)

The second equation is a lot more friendly because it can take advantage of properties of the difference of squares!

Try Some On Your Own The next page has 10 questions for you to try on your own.

Deb Russell