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#1 2007-04-04 11:35:26

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Exercises

(I tried to make it complicated, but then I realized it would be very hard : What I first made is
)

Last edited by Stanley_Marsh (2007-04-04 11:38:07)

Numbers are the essence of the Universe

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#2 2007-04-04 11:45:07

Stanley_Marsh
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Re: Exercises

Last edited by Stanley_Marsh (2007-04-04 11:46:51)

Numbers are the essence of the Universe

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#3 2007-04-04 11:52:06

Stanley_Marsh
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Re: Exercises

Last edited by Stanley_Marsh (2007-04-04 11:53:57)

Numbers are the essence of the Universe

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#4 2007-04-04 11:56:45

Stanley_Marsh
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Re: Exercises

Last edited by Stanley_Marsh (2007-04-04 11:58:11)

Numbers are the essence of the Universe

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#5 2007-04-04 12:12:30

Stanley_Marsh
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Re: Exercises

Last edited by Stanley_Marsh (2007-04-04 12:15:43)

Numbers are the essence of the Universe

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#6 2007-04-04 20:41:22

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Exercises

Testing this for the first few values of n, I find a[sub]1[/sub] = 2, a[sub]2[/sub] = 4, a[sub]3[/sub] = 6, a[sub]4[/sub] = 8, a[sub]5[/sub] = 10. So I suspect a[sub]n[/sub] = 2n.

Lets verify this by induction. Obviously a[sub]1[/sub] = 2 = 2(1) so its true for 1.

If its true for n, then

Hence, by induction, a[sub]n[/sub] = 2n for all n.

Last edited by JaneFairfax (2007-04-11 16:55:55)

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#7 2007-04-05 02:37:28

Stanley_Marsh
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Registered: 2006-12-13
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Re: Exercises

Induction is one method, you can also use

Last edited by Stanley_Marsh (2007-04-05 02:40:38)

Numbers are the essence of the Universe

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#8 2007-04-11 16:55:22

JaneFairfax
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Registered: 2007-02-23
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Re: Exercises

2.

Actually, I cheated slightly  I assumed beforehand that the series was absolutely convergent so I could take liberties with rearranging the summation terms.

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#9 2007-04-11 19:01:11

Stanley_Marsh
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Registered: 2006-12-13
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Re: Exercises

Your method works too, nice job.

Last edited by Stanley_Marsh (2007-04-11 19:06:00)

Numbers are the essence of the Universe

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#10 2007-04-11 20:22:03

JaneFairfax
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Registered: 2007-02-23
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Re: Exercises

Argh! I actually arrived at more or less the same final process as you, only I did it by a tedious route involving more work than necessary. Argh!

Last edited by JaneFairfax (2007-04-11 20:22:41)

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#11 2007-04-11 20:39:36

luca-deltodesco
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Registered: 2006-05-05
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Re: Exercises

Last edited by luca-deltodesco (2007-04-11 20:40:30)

The Beginning Of All Things To End.
The End Of All Things To Come.

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#12 2007-04-11 21:23:51

JaneFairfax
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Registered: 2007-02-23
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Re: Exercises

4.

Hmm

Could it be

Lets see.

so theyre fine.

Yes!

I can write the result in one single formula:

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#13 2007-04-12 02:38:42

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Exercises

You are right about both question.
About 3 you don't have to use root formula

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#14 2007-04-14 14:42:50

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Exercises

I am gonna give the answer to the last question.

Last edited by Stanley_Marsh (2007-04-14 18:02:15)

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#15 2007-04-28 06:08:09

catgirl
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Registered: 2007-03-21
Posts: 8

Re: Exercises

Whats so funny, JaneFairFax?

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