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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,058

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".

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,058

pls explain.

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 97,181

Here is a little hint:

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,188

hi niharika_kumar

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

Bob

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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: 97,181

Did you get the answer?

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,058

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: 97,181

Nope, it is rather easy! No computation necessary.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,058

pls. explain

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 97,181

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.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,058

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: 97,181

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.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,058

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: 97,181

Very good!

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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