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Sum

cinelli
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Sum

#371935

Postby cinelli » January 1st, 2021, 10:24 am

Evaluate the sum

.       1                 1                 1
.--------------- + --------------- + --------------- + ...
.sqrt(1)+sqrt(2) sqrt(2)+sqrt(3) sqrt(3)+sqrt(4)
.
. 1
. + ---------------------------
. sqrt(4088483)+sqrt(4088484)

Happy new year to all readers.

Cinelli

Gengulphus
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Re: Sum

#371958

Postby Gengulphus » January 1st, 2021, 10:53 am

cinelli wrote:Evaluate the sum

.       1                 1                 1
.--------------- + --------------- + --------------- + ...
.sqrt(1)+sqrt(2) sqrt(2)+sqrt(3) sqrt(3)+sqrt(4)
.
. 1
. + ---------------------------
. sqrt(4088483)+sqrt(4088484)

Happy new year to all readers.

Happy new year to you (and all others here as well) - but I'm afraid I'm about to 'spoil' it!

In general, (x+y)*(x-y) = x^2 - y^2. Substituting x = sqrt(N+1) and y = sqrt(N) gives (sqrt(N+1)+sqrt(N))*(sqrt(N+1)-sqrt(N)) = (sqrt(N+1))^2 - (sqrt(N))^2 = (N+1)-(N) = 1.

So 1/(sqrt(N)+sqrt(N+1)) = sqrt(N+1)-sqrt(N) = -sqrt(N)+sqrt(N+1). Applying that to each term of your sum changes it to -sqrt(1)+sqrt(2)-sqrt(2)+sqrt(3)-sqrt(3)+sqrt(4)-...-sqrt(4088483)+sqrt(4088484), in which almost all terms cancel, reducing it to just -sqrt(1)+sqrt(4088484) = -1+2022 = 2021.

Or of course there's the even quicker method of simply asking oneself "What else could the answer to a numeric puzzle posted today be?"... ;-)

Gengulphus

UncleEbenezer
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Re: Sum

#372084

Postby UncleEbenezer » January 1st, 2021, 3:39 pm

Gengulphus wrote:Happy new year to you (and all others here as well) - but I'm afraid I'm about to 'spoil' it!

The timestamp on that is less than half an hour after cinelli's post. You saw it, solved it, and posted here all in that time!

That you should solve it before me is to be expected. That you should do so before I've even seen it is ... um ... bloomin' cheek!

Happy new year 2022-1.

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Re: Sum

#372204

Postby mark88man » January 1st, 2021, 9:21 pm

Very clever - that took me back.

cinelli
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Re: Sum

#372219

Postby cinelli » January 1st, 2021, 10:52 pm

UncleEbenezer wrote:That you should solve it before me is to be expected. That you should do so before I've even seen it is ... um ... bloomin' cheek!

Bad luck, UncleE. I'm sure you would have got the right answer.

Cinelli

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Re: Sum

#372225

Postby UncleEbenezer » January 1st, 2021, 11:09 pm

cinelli wrote:
UncleEbenezer wrote:That you should solve it before me is to be expected. That you should do so before I've even seen it is ... um ... bloomin' cheek!

Bad luck, UncleE. I'm sure you would have got the right answer.

Cinelli

:D

I'd've needed a moment of clarity, which would come after reading it, then putting it aside and doing other things. Most likely to get the inspiration while (or straight after) out walking, and that's currently hopelessly constrained by the ice underfoot.

I'm not blaming him. Just mildly bemused. And the spoiler works perfectly: the solution wasn't thrust upon me unintentionally!

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Re: Sum

#372270

Postby Gengulphus » January 2nd, 2021, 9:32 am

UncleEbenezer wrote:
Gengulphus wrote:Happy new year to you (and all others here as well) - but I'm afraid I'm about to 'spoil' it!

The timestamp on that is less than half an hour after cinelli's post. You saw it, solved it, and posted here all in that time!

That you should solve it before me is to be expected. That you should do so before I've even seen it is ... um ... bloomin' cheek!

It works both ways. For example, both GoSeigen and you had the bloomin' cheek to solve viewtopic.php?f=73&t=26795 before I'd even seen it - both of you really ought to have had the decency to restrain yourselves while I was getting my computer set up in a new location! ;-)

But very quickly-posted solutions are basically a matter of luck - one needs all the ingredients to happen to come together at once, which they happened to do for me on this occasion:

* Happening to look for TLF posts to read shortly after the puzzle is posted (I think I saw this one about 20 minutes after cinelli posted it, whereas my average for seeing newly-posted puzzles must be many hours).

* The puzzle being quickly solvable - e.g. this one is solvable in less than a minute if one happens to think of the right approach (which I did), whereas all but the simplest of alphametics take quite a lot of time using the clues they provide to nibble away at the boundaries of one's ignorance about the solution until it's all gone and only knowledge of the solution is left.

* A post describing the solution being quickly writeable, which for many puzzles isn't the case. E.g. the knight's-tour puzzle in the above link can be solved quite quickly by printing it out and getting to work on the printout with a pen or pencil, but writing down the solution in a post takes considerably longer, especially if one wants to describe how one arrived at the answer rather than just giving the answer (writing my 'partial spoiler' post currently a bit over halfway down that thread took considerably longer than solving the puzzle, and extending it to a description of the full solution would probably have increased that time three- or four-fold).

Gengulphus


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