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### Find the 10th Number

```
Date: 08/10/2001 at 18:37:56
Subject: Patterns

I was given the following pattern and was asked to find the 10th
number in the sequence:

2 4 3 6 5 10

2 4 3 6 5 10 9 15 14 25

My teacher said the 25 was wrong. She said the answer was 36. She
multiplied by 2, and then subtracted 1. I calculated the pattern and
found 15 by doing the following:

2+3 = 5; 5+9 = 14; and 4+6 = 10; 10+15 = 25

She also said there was no logic to the 15. I found that every first
and third equalled the fifth and then every second and fourth number
equalled the sixth number. Starting with 4, the number following every
other number decreased by one and then increased.  That's where my 15
came from.  Who's right?
```

```
Date: 08/11/2001 at 23:37:47
From: Doctor Peterson
Subject: Re: Patterns

It's entirely possible for a sequence problem like this to have more
than one reasonable answer, since it's all a matter of guessing what
valid.

Yes, you can get 5 from 2 and 3 following your pattern, and 14 from 5
and 9; but where did the 9 come from? You haven't explained that,
other than by a vague idea that it has to increase. I think you
guessed.

You say the pattern is that the first plus the third equals the fifth.
If we follow that pattern, then we get

2  +  3  = 5
4  +  6 = 10
3  +  5  =  8
6  + 10  = 16
5  +  8  = 13
10  + 16  = 26

2, 4, 3, 6, 5,10, 8,16,13,26
-- -- -- --

So the tenth number is 26. This would be a perfectly valid answer,
since it takes a pattern that does exist in all the given numbers, and
continues it. Your numbers don't fit the pattern you identified,
because 3+5 is not 9.

So if you had just followed your own pattern consistently, you would

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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