Consecutive Integers

Date: 04/16/97 at 14:24:23
From: JARED ROSS
Subject: Algebra

If two consecutive numbers are less than one hundred, what is the 
larger number?

Date: 06/25/97 at 17:10:03
From: Doctor Sydney
Subject: Re: Algebra

Dear Jared,

Hello!  Thanks for writing. In reading your question, we came up with 
a couple of different ways to interpret it, so we aren't quite sure 
what you mean.  One way we might intepret the question is the 
following:

1. Suppose we have two consecutive numbers, each of which is less 
   than 100. What is the larger number?  

Or, you might mean:

2. Suppose that we have two consecutive numbers such that their sum 
   is less than 100.  Then what is the larger number?

This second question is more difficult than the first, so we'll assume 
that that is what you mean. However, if you were really asking the 
first question and still need help with it, write back, and we can 
help out.

So, let's look at the second interpretation.  Well, there are many 
pairs of numbers that when added up, equal less than 100.

Think of it this way: If the two numbers must add up to be less than 
100 (and we are only dealing with integers, which are numbers of the 
type ...,-3, -2, -1, 0, 1, 2, 3,...),  then we can express what we 
want as:

      n + (n + 1) < 100

This says that n, plus the number that comes right after n, must be 
equal to a number less than 100. But we can make this even simpler! 
The equation can also be written as:

           2n + 1 < 100 
               2n < 99
                n < 49.5 

So, for any pair of consecutive numbers (n, n+1) such that n < 49.5, 
the sum of the consecutive numbers is less than 100. For instance, the 
consecutive numbers (49,50) add up to less than 100.  In fact 49 and 
50 are the largest consecutive numbers that have this property. See if 
you can figure out why. Can you find other examples of consecutive 
numbers that work?  

I hope this helps you. Please write back if we answered the wrong 
question or if you need more help.

-Doctors Matthew and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/