Algebra problems often ask about properties of consecutive numbers, such as 1, 2, 3 or 9, 10, 11.

The numbers 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.

If 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 of Consecutive Numbers

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

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!

### Consecutive Number Questions

Try these consecutive number problems on your own. Even if you can figure some of them out without the methods above, try them using consecutive variables for practice:

1. Four consecutive even numbers have a sum of 92. What are the numbers?

2. Five consecutive numbers have a sum of zero. What are the numbers?

3. Two consecutive odd numbers have a product of 35. What are the numbers?

4. Three consecutive multiples of five have a sum of 75. What are the numbers?

5. The product of two consecutive numbers is 12. What are the numbers?

6. The sum of 4 consecutive integers is 46. What are the numbers?

7. The sum of 5 consecutive even integers is 50. What are the numbers?

8. If you subtract from the product of 2 consecutive numbers, the sum of the same two numbers, the answer is 5.

What are the numbers?

9. Do there exist 2 consecutive odd numbers with product 52?

10. Do there exist 7 consecutive integers with sum of 130?

### Solutions

1. 20, 22, 24, 26

2. -2, -1, 0, 1, 2

3. 5, 7

4. 20, 25, 30

5. 3, 4

6. 10, 11, 12, 13

7. 6, 8, 10, 12, 14

8. -2 and -1 OR 3 and 4

9. No. Setting up equations and solving leads to a non integer solution for x.

10. No. Setting up equations and solving leads to a non integer solution for x.