Associated Topics || Dr. Math Home || Search Dr. Math

### Summing Three Consecutive Numbers

```
Date: 12/03/98 at 17:30:32
From: MEAGAN
Subject: Consecutive numbers

The problem I am working with is:

The number 6 can be expressed as the sum of three consecutive numbers.
How many other natural numbers from 10 to 40 can also be written as a
sum of three consecutive natural numbers?

I don't understand any of the problem. I was hoping you could take me
step by step.
```

```
Date: 12/04/98 at 10:33:15
From: Doctor White
Subject: Re: Consecutive numbers

Meagan:

Let's look at some definitions first.

1) Natural numbers are the numbers that we count with (1,2,3,4,5,...).
2) Consecutive numbers mean in a row, such as 1,2,3 or 15,16,17, etc.
3) Sum means to add together.

Now: 6 can be expressed as the sum of three consecutive numbers because
6 = 1 + 2 + 3. Another example: 15 + 16 + 17 = 48. So, 48 is the sum of
three consecutive numbers.

So, one way to solve your problem is to start checking sets of three
consecutive numbers.

Another way is through the use of algebra.

Let X be the first of three consecutive numbers. Then the second of the
three consecutive numbers must be X + 1. Also, the last of the three
must be X + 2.

Here is a table to show this:

X        X + 1           X + 2            Sum

1        (1+1) = 2       (1+2) = 3         6

7        (7+1) = 8       (7+2) = 9        24

From here we can set up equations to test for your answers:

X + X+1 + X+2 = 10
3X + 3 = 10
3X = 7
X = 2 1/3

Since X is not an integer, there are not three consecutive natural
numbers that add up to 10.

Another example:

X + X+1 + X+2 = 24
3X + 3 = 24
3X = 21
X =  7

So, 7 + (7+1) + (7+2) = 7 + 8 + 9 = 24. There are three consecutive
natural numbers that add up to 24.

Hope this has helped you to see this problem more clearly. If you need
further help, let us know.

- Doctor White, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra