Equation True for Any Three Consecutive Numbers

Date: 02/14/2003 at 00:13:46
From: Beverly
Subject: Word problems

Find three consecutive odd numbers, such that 6 less than 2 times the 
larger number is equal to the sum of the other two numbers.  

Please help!

I came up with:  2(x+4)-6 = x + x+2
I always end up with 2x + 2 = 2x + 2

Date: 02/14/2003 at 11:22:00
From: Doctor Ian
Subject: Re: Word problems

Hi Beverly,

Thanks for showing your work. That makes it easier to know where to
start.  

I get the same thing you did.  Let's take it a step farther, though:

  2(x+4) - 6 = x + x + 2

  2x + 8 - 6 = 2x + 2

      2x + 2 = 2x + 2

           2 = 2

What does this mean? It's a true statement, and there's no x in sight.  
What that means is that the truth of the original equation doesn't 
depend on x. That is, it's true for _any_ three consecutive numbers. 

Let's try some, as a check:

  1) 3, 5, 7

     6 less than 2 times 7 equals 3 plus 8.
     6 less than 14 equals 8. 
     8 equals 8.
     True!

  2) 11, 13, 15

     6 less than 2 times 15 equals 11 plus 13. 
     6 less than 30 equals 24.
     24 equals 24.
     True!

So there are at least two cases where it works. If you check others,
you'll find that they work as well. 

Okay, can we see _why_ it works? Well, if N is any integer, then 
(2N+1) is an odd integer. (Do you see why?) So we can represent three 
consecutive odd integers this way:

  (2N+1), (2N+3), (2N+5)

Now, let's add the first two. We get

  2N + 1 + 2N + 3 = 4N + 4

Now let's see what 6 less than 2 times the largest is:

  2(2N+5) - 6 = 4N + 10 - 6

              = 4N + 4

So no matter _which_ three consecutive odd numbers we start out with,
this condition will always be true. And this is what the 'solution'

  2 = 2

tells us.  

It's interesting to compare this with a case where _no_ value of the
variable will work. For example, suppose I ask you to find three
consecutive numbers such that the sum of the first two is equal to
twice the third plus 2.  

A little thought will show that the problem can't possibly have a
solution. Why? Because the sum of the first two, being the sum of two 
odd numbers, will have to be even. But the third number is also odd, 
so adding 2 to it will give us another odd number. And we can't have 
an even number be equal to an odd number. 

Now, what happens if we try to 'solve' it? We get

  x + x + 2 = 2(x+4) + 2

     2x + 2 = 2x + 8 + 2

     2x + 2 = 2x + 10

          2 = 10

which, of course, is false.  

So these are two cases that can come up sometimes when you're trying
to solve for a variable in an equation:  

  (1) the variable disappears, leaving a true statement, 
      which means that _any_ value will satisfy the original 
      equation, or 

  (2) the variable disappears, leaving a false statement, 
      which means that _no_ value will satisfy the original 
      equation.  

Of course, if you think you've found one of these cases, it's always a
good idea to go back over your calculations to be sure you didn't drop
a sign, or make an arithmetic error somewhere. But the important
thing is to know that these cases _can_ arise, and what they mean when
they do. 

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/