Sum of powers of 2

lepton · **Joined:** Sun, 6 Apr 2008 23:49:57 UTC **Posts:** 143

Is there a closed formula for the first n powers of an integer? In particular, is there a closed formula for $2^1+\cdots+2^n$ ?

helmut · **Posted:** Fri, 22 Jan 2010 02:27:04 UTC

Matt · **Joined:** Wed, 1 Oct 2003 04:45:43 UTC **Posts:** 9642

It's actually $2^{n+1}-1$

helmut · **Posted:** Fri, 22 Jan 2010 09:02:49 UTC

No, Matt. Your formula would be correct if the sum started with 2^0

.

Soroban · **Posted:** Fri, 22 Jan 2010 16:25:11 UTC

Hello, lepton!

Quote:

Is there a closed formula for the first n powers of an integer?
In particular, is there a closed formula for $2^1+2^2 +\cdots+2^n$ ?

Yes, you have a geometric series
. . with first term a = 2

, common ratio r = 2

, and

terms.

The sum is: . $$S \;=\;a\,\frac{1-r^n}{1-r}$

So we have: . $$2\,\frac{1-2^n}{1-2}\;=\;2(2^n-1)$ . . . as helmut pointed out.

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

You could derive the formula with a little acrobatics.

$\begin{array}{ccccc}\text{We have:} & S&=& 2 + 2^2 + 2^3 + 2^4 + \hdots + 2^n \qquad & [1] \\ \text{Multiply by 2:} & 2S &=& \qquad\quad 2^2 + 2^3 + 2^4 + \hdots + 2^n + 2^{n+1} & [2]\\\end{array}$

Subtract [1] from [2]: . $S \;=\;2^{n+1} - 2$

Matt · **Joined:** Wed, 1 Oct 2003 04:45:43 UTC **Posts:** 9642

helmut wrote:

No, Matt. Your formula would be correct if the sum started with 2^0

.

Indeed you are right. My mistake.

Soroban wrote:

The sum is: . $$S \;=\;a\,\frac{1-r^n}{1-r}$

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

lepton · **Joined:** Sun, 6 Apr 2008 23:49:57 UTC **Posts:** 143

Sweet, thank you very much everybody.

Shadow · **Posted:** Mon, 25 Jan 2010 17:48:20 UTC

Matt wrote:

helmut wrote:

No, Matt. Your formula would be correct if the sum started with 2^0

.

Indeed you are right. My mistake.

Soroban wrote:

The sum is: . $$S \;=\;a\,\frac{1-r^n}{1-r}$

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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Sum of powers of 2

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