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/