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.
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 $\displaystyle S_n = \frac{a(r^n - 1)}{r - 1}$
$\displaystyle = \frac{2(3^n - 1)}{3 - 1}$
$\displaystyle = \frac{2(3^n - 1)}{2}$
$\displaystyle = 3^n - 1$.