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Wrong Number in a Sequence

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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

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/
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Associated Topics:
High School Sequences, Series

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