Originally posted by JS357
Would it be Ok to approach this collaboratively, suggesting ideas?
If so, one idea I have had is that if the sum of 1 to (M-1) (I'll call it the LHS sum) is equal to the sum of M+1 to N (the RHS sum), then the quantity ((RHS sum) - (M*N)) is equal to summation of 1 to (N-M). It subtracts M from each term on the RHS.
Even if the above is correct and on t ...[text shortened]... M from the LHS brings the two sides into equality: sum (1 to M-1) - (M*N) = sum (1 to N-M). So?
"Would it be OK to approach this collaboratively, suggesting ideas?"
When I created this thread, my aim was to encourage people who normally
don't have to do mathematics to have some fun in attempting something unusual
(in stepping outside their 'comfort zone'
. If you feel that you could still enjoy
solving a problem collaboratively, I don't mind. Again, I would just ask that you
don't write a computer programme because that's against the spirit of the problem.
Of course, I already know how to solve these problems (which I first did on my
own as a young student). Given that I don't wish to appear too much like a nanny
here, I would prefer not to oversupervise you. I would like to encourage you to
take some steps on your own even at the risk of falling down. You learn maths
by attempting to do it, not just by reading about someone else has done it.
You learn chess by playing, not just by reading books about how to play chess.
Often, when I have explained how to solve a problem to someone who had been
perplexed by it, I have been told, "It's all so simple! Surely what you did must
be trivial. I must have known how to solve it all along; I just didn't express it."
For whatever it's worth, I don't come from a mathematically-inclined family;
my parents had little education in mathematics. Initially, I was stereotyped at
school as extremely poor at mathematics. A teacher said that I obviously was
hopeless, and the children from more educated and privileged families liked to
put me down. That was because I never had been taught anything about even
basic arithmetic. So one day my father finally decided to teach me how to add,
subtract, multiply, and divide--which I learned that day (it took a few hours).
He demanded that I be able to perform basic arithmetic almost instantly and
always correctly. He would drill me by shouting, for example, "What's 13 x 13?",
and I would reply, '169'. As usual, the implicit threat of a beating hung over me.
Most middle-to-upper class Western children, I suppose, would have found the
"throw me into the deep end and sink-or-swim" conditions unbearable. But I
was a natural swimmer. When school resumed, everyone else seemed shocked
that I could perform basic arithmetic more quickly and accurately than every
other student and my 'teacher'. Of course, then there came strong pressure from
the school authorities for me to conceal my intelligence as much as possible.
Someone from my background was expected to stay near the bottom, not to
surpass the children from much more privileged and supportive families.
Those children had more opportunities and diversions (e.g. piano lessons), so
they never had to seek fulfillment in the rather bleak circumstances around me
and they presumably never shared my delight in drawing much from little.
Personally, I could not understand why everyone else did not find 'school'
mathematics as effortless as I did. After all, the basic facts of arithmetic (e.g.
the multiplication tables) are well-known and invariant, so why should anyone
ever make any errors? So I did not consider myself as anyone special at maths.
Later, I was brought to the attention of some mathematicians, who told me that
I obviously was extraordinary and who attempted to convince my parents of it.
Of course, my parents found it extremely hard to believe that I could ever be
better than ordinary, if that, in doing anything. In contrast to Ruth Lawrence,
for instance, I was a rather indolent and unambitious student. A few years after
I learned arithmetic, I learned number theory (at an undergraduate level), but
I had no one around me with whom to communicate my mathematical thoughts.
So my isolated circumstances hindered my development. Ramanujan thrived
in isolation, but, of course, I could hardly approach his level of natural genius.