What You Need to Know About Consecutive Numbers

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The concept of consecutive numbers may seem straightforward, but if you search the internet, you'll find slightly differing views about what this term means. Consecutive numbers are numbers that follow each other in order from smallest to largest, in regular counting order, notes Study.com. Put another way, consecutive numbers are numbers that follow each other in order, without gaps, from smallest to largest, according to MathIsFun.

And Wolfram MathWorld notes:

"Consecutive numbers (or more properly, consecutive ​integers) are integers n1 and n2 such that n2–n1 = 1 such that n2 follows immediately after n1.​"

Algebra problems often ask about properties of consecutive odd or even numbers, or consecutive numbers that increase by multiples of three, such as 3, 6, 9, 12. Learning about consecutive numbers, then, is a bit trickier than is at first apparent. Yet it is an important concept to understand in math, particularly in algebra.

Consecutive Number Basics

The numbers 3, 6, 9 are not consecutive numbers, but they are consecutive multiples of 3, which means that the numbers are adjacent integers. A problem may ask about consecutive even numbers—2, 4, 6, 8, 10—or consecutive odd numbers—13, 15, 17—where you take one even number and then the very next even number after that or one odd number and the very next odd number.

To represent consecutive numbers algebraically, let one of the numbers be x.

Then the next consecutive numbers would be x + 1, x + 2, and x + 3.

If the question calls for consecutive even numbers, you would have to ensure that the first number you choose is even. You can do this by letting the first number be 2x instead of x. Take care when selecting the next consecutive even number, though.

It is not 2x + 1 since that would not be an even number. Instead, your next even numbers would be 2x + 2, 2x + 4, and 2x + 6. Similarly, consecutive odd numbers would take the form: 2x + 1, 2x + 3, and 2x + 5.

Examples of Consecutive Numbers

Suppose the sum of two consecutive numbers is 13. What are the numbers? To solve the problem, 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, your numbers are 6 and 7.

An Alternate Calculation

Suppose you had chosen your consecutive numbers differently from the start. In that case, 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, as follows:

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

Here you find that x equals 10, while in the previous problem, x was equal to 6. To clear up this seeming discrepancy, substitute 10 for x, as follows:

  • 10 - 3 = 7
  • 10 - 4 = 6

You then have the same answer as in the previous problem.

Sometimes it may be easier if you choose different variables for your consecutive numbers. For example, if you had a problem involving the product of five consecutive numbers, you could calculate it using either of the following two methods:

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

or

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

The second equation is easier to calculate, however, because it can take advantage of properties of the difference of squares.

Consecutive Number Questions

Try these consecutive number problems. Even if you can figure out some of them without the methods discussed previously, 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. If the sum of four consecutive integers is 46, what are the numbers?

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

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

9. Do there exist two consecutive odd numbers with a product of 52?

10. Do there exist seven consecutive integers with a 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.

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Russell, Deb. "What You Need to Know About Consecutive Numbers." ThoughtCo, May. 6, 2018, thoughtco.com/properties-of-consecutive-numbers-2311939. Russell, Deb. (2018, May 6). What You Need to Know About Consecutive Numbers. Retrieved from https://www.thoughtco.com/properties-of-consecutive-numbers-2311939 Russell, Deb. "What You Need to Know About Consecutive Numbers." ThoughtCo. https://www.thoughtco.com/properties-of-consecutive-numbers-2311939 (accessed May 22, 2018).