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Find the 10th Number


Date: 08/10/2001 at 18:37:56
From: Adam Fierman
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

I answered:

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

Hi, Adam.

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 
pattern the writer had in mind. But I don't see your answer as being 
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 
have a valid answer different from your teacher!

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

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