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 Post subject: Sum of powers of 2Posted: Fri, 22 Jan 2010 02:23:07 UTC
 S.O.S. Oldtimer

Joined: Sun, 6 Apr 2008 23:49:57 UTC
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Is there a closed formula for the first n powers of an integer? In particular, is there a closed formula for ?

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 Post subject: Posted: Fri, 22 Jan 2010 02:27:04 UTC

Joined: Sat, 26 Apr 2003 22:14:40 UTC
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Location: El Paso TX (USA)

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 Post subject: Posted: Fri, 22 Jan 2010 07:14:56 UTC
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It's actually

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 Post subject: Posted: Fri, 22 Jan 2010 09:02:49 UTC

Joined: Sat, 26 Apr 2003 22:14:40 UTC
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Location: El Paso TX (USA)
No, Matt. Your formula would be correct if the sum started with .

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 Post subject: Re: Sum of powers of 2Posted: Fri, 22 Jan 2010 16:25:11 UTC
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Location: Lexington, MA
Hello, lepton!

Quote:
Is there a closed formula for the first n powers of an integer?
In particular, is there a closed formula for
?

Yes, you have a geometric series
. . with first term , common ratio , and terms.

The sum is: .

So we have: . . . . as helmut pointed out.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

You could derive the formula with a little acrobatics.

Subtract [1] from [2]: .

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 Post subject: Re: Sum of powers of 2Posted: Sat, 23 Jan 2010 06:00:20 UTC
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helmut wrote:
No, Matt. Your formula would be correct if the sum started with .

Indeed you are right. My mistake.

Soroban wrote:
The sum is: .

This formula is not at all natural. It's easier to think of sum in terms of binary numbers...

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 Post subject: Posted: Sat, 23 Jan 2010 23:38:44 UTC
 S.O.S. Oldtimer

Joined: Sun, 6 Apr 2008 23:49:57 UTC
Posts: 143
Sweet, thank you very much everybody.

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 Post subject: Re: Sum of powers of 2Posted: Mon, 25 Jan 2010 17:48:20 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
Matt wrote:
helmut wrote:
No, Matt. Your formula would be correct if the sum started with .

Indeed you are right. My mistake.

Soroban wrote:
The sum is: .

This formula is not at all natural. It's easier to think of sum in terms of binary numbers...

or -ary numbers for the general case.

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