Wrong Number in a Sequence

Date: 08/13/99 at 15:39:13
From: Samantha Bossingham
Subject: Series question

I came across this question on a test for management. The question is 
as follows:

In the following sequence, one number is incorrect in the sequence. 
Which one is wrong, and what number does it need to be replaced with? 
Can you explain this sequence?

     1, 1, 2, 2, 6, 18, 21, 84, 88, ...

The choices were:
   a. 3
   b. 4
   c. 9 
   d. 24
   e. 48

What is the answer? I tried going through multiplication and division, 
adding and subtracting... I can't figure it out. Help!

Date: 08/21/99 at 01:36:29
From: Doctor Douglas
Subject: Re: series question

Hi, Samantha.

This is a very interesting puzzle. Here's the way I would look at it:

In the sequence {1, 1, 2, 2, 6, 18, 21, 84, 88, ...} I would first 
focus on the bigger numbers, since whatever rule governs the series 
should be easier to spot there. There are lots of plausible 
connections between small numbers such as 1 and 2 and 6. To sort it 
out from all these possibilities seems very difficult. But at the 
larger numbers, there are fewer plausible relationships. Indeed, that 
21 and 84 seem to be connected by the operation (multiply-by-four). 
And then 84 to 88 seems to be accomplished by the simple operation 
(add-four).

So we could write things like this:

     1    1    2    2    6    18    21    84    88
                                      (*4)  (+4)

Hmmm, now how do we connect 18 and 21? Perhaps the simplest is to say 
that we need to add 3:

     1    1    2    2    6    18    21    84    88
                          (*3)  (+3)  (*4)  (+4)

I've also added the (multiply-by-3) that seems to connect 6 to 18. Now 
perhaps you can see a pattern developing. Try exploring this and see 
if it leads anywhere toward helping identify the "wrong" number and 
what its "correct" value should be.

If you find this hint confusing, please don't hesitate to write back.

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