# Need help with Combinatoric Proof

• Feb 27th 2010, 09:38 PM
icefirekid
Need help with Combinatoric Proof
Hi I need help to find a combinatoric proof regarding :

$2.3^0+2.3^1+...+2.3^{n-1}=3^n -1$

Thanks in advance for the help.
• Feb 27th 2010, 10:21 PM
Prove It
Quote:

Originally Posted by icefirekid
Hi I need help to find a combinatoric proof regarding :

$2.3^0+2.3^1+...+2.3^{n-1}=3^n -1$

Thanks in advance for the help.

I take it that the dots are being used to represent multiplication (though if so they should be written higher, using the command \cdot).

Notice that this is a geometric series, with $a = 2, r = 3$.

So $S_n = \frac{a(r^n - 1)}{r - 1}$

$= \frac{2(3^n - 1)}{3 - 1}$

$= \frac{2(3^n - 1)}{2}$

$= 3^n - 1$.