Mike Raiford <"m[raiford]!at"@gmail.com> wrote:
> http://www.maa.org/devlin/LockhartsLament.pdf

  That reminds me vividly of something which I have noticed and realized
all by myself: I have never seen *anywhere* the *logical* explanation of
how you can easily calculate, for example, the sum of all the integers
from 1 to 100. It's *always* just the raw and sterile formula, and that's
it. No explanation, no intuitive nor logical way of deducing it. Just the
formula and that's it.

  The raw formula is rather useless by itself when you don't understand
where it's coming from. If you change the problem slightly, for example
to "sum of all integers from 1 to 200", most people will still be able
to *guess* how to change the formula to get the result (ie. you just
change the '100' to a '200'). However, change it eg. to "all the even
integers between 50 and 250" and most people are left completely baffled.
No idea whatsoever. Why? Because they don't understand where the formula
is coming from.

  A more logical approach to actually deducing the formula is to think
that summing all the integers between 1 and 100 is the same as summing
the average of those numbers with itself as many times as there are
numbers in that range. (In other words, the product of the average and
the amount of numbers, ie. 100 in this case.) This can be intuitively
demonstrated graphically (if you visualize the integers as vertical
bars of the length of the integer, forming a triangle construct, you
can split this construct in half and mirror the upper half to get a
rectangle).

  Knowing this makes it easy to deduce the answer to a whole lot of other
similar problems. For example "sum of all the even integers between 50
and 250" can be done with the same principle: The average of all those
numbers times their amount. (The average is, obviously, (50+250)/2, and
the amount of values is (250-50)/2 + 1.)

  (Of course you need to be careful to not to apply this idea too far.
For example it cannot be applied to "sum of all squared numbers between
1 and 100". If you visualize it graphically like earlier, you'll see why.)

-- 
                                                          - Warp

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