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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,021

pls tell me is there any easier way to solve the following:-

friendship is tan 90°.

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi,

Think "partial fractions" and "telescoping series".

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,021

pls explain.

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,772

Here is a little hint:

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,395

hi niharika_kumar

You'll find every fraction can be split like this and then you can do a lot of cancelling out.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,772

Did you get the answer?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,021

and if there is a long series, then is it necessary to compute each one?

don't you think it will be time consuming.

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,772

Nope, it is rather easy! No computation necessary.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,021

pls. explain

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,772

Hi;

Now subtract the bottom one from the top one. The technique is called telescoping. Notice the massive cancellations making the sum to any amount of terms easy.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,021

so here the answer is

.right?its a nice method but if we have just one equation like

what do we need to do now?

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,772

Hi;

When you have a series you must guess at the general term. You can play spot the pattern or you can use curve fitting but you will come up with

just as Nehushtan did above. You break it into partial fractions as above:

Now you telescope it out

Now just subtract the bottom from the top, again with massive cancellations.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,021

thanks again!

I initially couldn't understand it clearly.

thanks again.....

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,772

Very good!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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