Consecutive Numbers

Date: 12/13/97 at 16:50:18
From: Justin Noah
Subject: Consecutive numbers

I don't understand what consecutive numbers are. Can you please 
explain this to me ?

Thank you,
Justin Noah

Date: 12/16/97 at 11:27:55
From: Doctor Mark
Subject: Re: Consecutive numbers

Hi Justin,

Consecutive numbers are just numbers that sort of "lie next to each 
other.  

For instance, 2, 3, 4 are consecutive numbers, as are 131, 132, 133.  
Another way of saying this is that consecutive numbers are integers 
that you get by starting somewhere and then keeping on adding 1.  
So five consecutive numbers starting at 24 would be 24, 24+1 = 25, 
25+1 = 26, 26+1 = 27, 27+1 = 28, so the five consecutive numbers are 
24, 25, 26, 27, 28.

Slightly more complicated are consecutive *even* numbers. These are 
numbers that are even numbers, and "lie next to each other" in the 
only way that they can: like 2, 4, 6, or 154, 156, 158, 160. That is,  
to get consecutive even numbers, pick some even number, then add 2 
to it, then add 2 to *that*, and so on. For instance, four consecutive 
even numbers starting at 88 would be 88, 88+2 = 90, 90+2 = 92, 
92 + 2 = 94, so the consecutive even numbers are 88, 90, 92, 94.  
Said another way, start at some even number, then count by 2s:  88, 
90, 92, 94.

Consecutive *odd* numbers are similar, except that you start at an 
*odd* number, then keep adding 2. So if you want three consecutive odd 
numbers starting at 17, you keep adding 2 until you have your three 
numbers:  17, 17+2 = 19, 19+2 = 21; so the three consecutive odd 
numbers are 17, 19, 21.

This idea sometimes gets a little confusing when you start doing 
algebra, but you don't say whether that is what bothering you.  I'll 
answer it anyway.

When  you want to represent consecutive numbers in algebra, you start 
with the beginning number, say x, then keep adding 1 to it to get

   x, 

   x + 1

   (x+1) + 1 = x + 2

   (x+2) + 1 = x + 3

and so on, so consecutive numbers can be represented as x, x+1, x+2, 
and so on.

To represent consecutive even or odd numbers, you start somewhere, 
then add 2, then add 2 more, and so on:

   x

   x + 2

   (x+2) + 2 = x + 4

   (x+4) + 2 = x + 6

and so on.  So consecutive even *or* odd numbers are represented as x, 
x+2, x+4, x+6, and so on.

Note that consecutive even *or* odd numbers are represented the *same 
way* algebraically. That sounds confusing until you remember that if 
you are looking for consecutive even numbers, the numbers must all be 
even, so that means that you must start at an even number, so that x 
must be even. If you are looking for consecutive *odd* integers, then 
you have to pick x as an *odd* integer.  (This may take a while to 
understand...)

Hope this helps, and write back if you have any other questions.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/