A rather bright but troublesome student was giving his math teacher a difficult time. To keep him
occupied for what she thought would be a lengthy period, she assigned him the problem of adding
together all the numbers from 1 to 100. Much to her dismay, he came up with the solution in a
matter of seconds. How did he do it?
remember what I said about me & maths earlier? This would be where
I showed off my amazing abilty to tap many numbers into a calculator
very quickly.
I can see how both of these answers are essentially the same, and
that they give the correct answer. Is it possible to explain to me why?
Jon,
The student was asked to add up the first 100 integers
1 + 2 + 4 + ... + 97 + 98 + 99 + 100
See what happens when you write the same sequence directly
underneath:-
1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100
100 + 99 + 98 + 97 + ... + 4 + 3 + 2 + 1
Instead of adding up the rows, add up each individual column. In
other words, 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 and so on all
the way up to 99 + 2 = 101, and finally 100 + 1 = 101
Clearly, every "column" equals 101. The cunning bit I guess is that we
know how many columns there are. There must be 100 (since we've
written out 1, 2, 3, etc up to 100).
100 lots of 101 equals 10,100. So far so good. But why is the answer
5050? Remember that we added up the columns. Each column
contained 2 rows, that is we added up the first 100 integers *twice*.
Therefore we need to divide the answer by 2.
We had 100 lots of 101 and needed to divide by 2.
(100x101)/2 is the same (written slightly differently) as what Gilbert
and Hotpawn wrote.
Gauss realised all this in his head in a few seconds. The other children
would have got as far as 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 and so on,
adding each number in turn to the total of the previous numbers
added together. This is cumbersome and heavy handed.
As is so often in Mathematics, it wasn't that Gauss was brilliant at
adding up each number in turn, but rather he found an alternative,
simplified, more beautiful way of solving the problem.
Godfrey Hardy (might have been someone else) said "there is no
place for ugly mathematics".
Mark
The Squirrel Lover
It can be generalised to any arithmetic series (a sum in which the difference between one
term and the next is constant), so yes it is very useful. A similar technique can be used to
evaluate many other kinds of sums: you express them in the form
(a-b) + (b-c) + (c-d) + ..... + (y-z), and everything cancels except for a - z, which is your
answer. Nice.
It's a great story.
The student in question was Gauss. Historians disagree when he
actually did it but most concur that it was sometime between the ages
of 7 and 10 (probably aged 8) around about 1785. This paved the way
for the formalised theory of Arithmetic Progressions and Geometric
Grogressions.
I remember reading somewhere that what really transformed him from
precocious youngster to establised mathematician however came when
he was 19. He proved that a regular polygon of 17 sides could be
constructed with a compass and a straight edge. 2,000 years previous
the Greeks were obsessed about compass and straight edge
constructions.
Euclid had given constructions for regular polygons with 3, 4, 5, 6, 8,
10, 12, 16, 20, 32, 40, 48, 64 (and so on) but 17 proved quite tricky
for 2,000 years until Gauss came along (with help from work done by
Fermat).
For anyone interested in reading about Gauss then a good place to
look at is
http://www.geocities.com/RainForest/Vines/2977/gauss/gauss.html