Sequence Differences

Date: 06/24/2003 at 16:22:38
From: Anthony
Subject: Problem solving

The third and fourth terms of a sequence are 26 and 40. If the second 
differences are a constant 4, what are the first five terms of the 
sequence?

Date: 06/25/2003 at 08:20:22
From: Doctor Edwin
Subject: Re: Problem solving

Hi, Anthony.

I'm not sure where your difficulty is. Is it that you don't know 
what "second differences" means, or that you don't know how to apply 
it to this problem?

Let's talk about the terminology for a minute. "First difference" is 
the difference between the numbers in the series. The series itself 
looks like:

  ?   ?   26  40  ?

That is, we know two out of the five terms. So what's the first 
difference between the third and fourth terms?

            14      -- first difference
           / \    
  ?   ?   26  40  ? -- series

Now the second difference is the difference between the first 
differences. They've told us that it's 4 for all the terms:


      4   4   4 -- second difference
     / \ / \ / \
    ?   ?  14   ? -- first difference
   / \ / \ / \ / \
  ?   ?   26  40  ? -- series


So what's the last first difference in our picture? It's going to be 
4 more than the one before it, or 18:


      4   4   4 -- second difference
     / \ / \ / \
    ?   ?  14  18 -- first difference
   / \ / \ / \ / \
  ?   ?   26  40  ? -- series

Which means that the last number in the series is 18 more than the 
one before it. Is that enough to get you started? Write back if you 
need more help.

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