by Kurt » Tue Apr 17, 2007 9:28 pm
What does APMO stand for? I've never heard of that one (or almost any of the ones that have popped up in the forums today).
Anyway, I'll give the problem a shot:
You have the sum of the reciprocal of triangular numbers:
Since [unparseable or potentially dangerous latex formula]
Decomposing into partial fractions, [unparseable or potentially dangerous latex formula]
So [unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]. It's obvious that the inner sums will cancel with each other, and so all that is left is: [unparseable or potentially dangerous latex formula] for the sum of k triangular inverse numbers [unparseable or potentially dangerous latex formula]
So [unparseable or potentially dangerous latex formula].
And so [unparseable or potentially dangerous latex formula].
Which is [unparseable or potentially dangerous latex formula].
Now using a handy trick for powers of 2 and reciprocals: [unparseable or potentially dangerous latex formula].
This is true by AM>HM, (or in this case, 1/HM > 1/AM)
We have 9 such cases here (from [unparseable or potentially dangerous latex formula]to [unparseable or potentially dangerous latex formula])
And of course we have a couple of terms before that ([unparseable or potentially dangerous latex formula]).
So total we have [unparseable or potentially dangerous latex formula].
However, for the part of the sum that is the sum of the reciprocals, that is just 1024 of the 1996 terms...So it is obviously greater than 1001.
Last edited by
Kurt on Tue Apr 17, 2007 9:50 pm, edited 3 times in total.