Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 GARFIELD 20/11/84; site garfield.UUCP
Path: utzoo!utcsri!ubc-vision!garfield!robertj
From: robertj@garfield.UUCP
Newsgroups: net.math
Subject: Counter-Intuitive Sequences
Message-ID: <748@garfield.UUCP>
Date: Fri, 31-Jan-86 09:32:59 EST
Article-I.D.: garfield.748
Posted: Fri Jan 31 09:32:59 1986
Date-Received: Fri, 31-Jan-86 12:43:23 EST
Sender: paulc@garfield.UUCP
Reply-To: robertj@garfield.UUCP (Robert Janes)
Followup-To: net.math
Organization: Memorial U. of Nfld. C.S. Dept., St. John's
Lines: 56
Keywords: jumping to conclusions



	This posting regards the poster who asked the persons on net.math
to find the sum of the sequence

	1 + 3 + 6 + 10 + 15 + ....... +an

presumably this meant that he wanted us to find the sum of the sequence
defined recursively as:

	a1  = 1
	ak = a(k-1) + k   1<k<=n

and it was the sum 


		k=n
		_____	
		-
		 _   ak
		_
		______
		k = 1

which was desired.

	The tricky part of this though is not finding the sum but making
the leap of faith to the above given definition of the sequence ( this is
probably what the test wanted however ). The interesting thing here is that
there is no truly mathematical way to make this jump ( at least not that
I have heard of ) and I would like to exhibit a famous ( but hopefully not
too famous example of where the obvious breaks down ).

Consider the following sequence:

		1, 2, 4, 8, 16,....

what is the next term ?	

Invariably the response is "32".
This however does not have to be the case and an alternate sequence arises
very naturally. Consider the sequence of n where n is the number of regions
the interior of a circle can be divided into using k lines where k starts
at 0. If you do that for the first five k you get the above sequence.
Suggestive isn't it ? However it turns out that for k=6 n=31 and the 
intuitive result falls flat on its face.
The "correct" answer (in this sense) is "31".
However any IQ test in which you answered "31" you would probably be put
down as a low-grade moron then and there so when in doubt answer "32" !

	Any other interesting sequences with both intuitive and counter-
intuitive but "natural" continuations ( random sequences are not interesting
in other words, use your intuition as to what I mean ).


					Robert Janes