Hi I need help to find a combinatoric proof regarding :

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

Thanks in advance for the help.

Printable View

- Feb 27th 2010, 08:38 PMicefirekidNeed help with Combinatoric Proof
Hi I need help to find a combinatoric proof regarding :

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

Thanks in advance for the help. - Feb 27th 2010, 09:21 PMProve It
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 $\displaystyle a = 2, r = 3$.

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

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

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

$\displaystyle = 3^n - 1$.