Originally Posted by

**maddas** Well, write it all out.

$\displaystyle 3\cdot 4^2 + 3^2\cdot 4 + 3^3\cdot 1 + {3^4\over 4} + {3^5\over 4^2} + {3^6\over 4^3} +\cdots$

Now factor the later part like

$\displaystyle 3\cdot 4^2 + 3^2\cdot 4 + 3^3\Big(1 + {3\over 4} + {3^2\over 4^2} + {3^3\over 4^3} +\cdots\Big)$

The sum in parenthesis is geometric. You should know that 1+r+r^2+...=1/(1-r) for |r|<1. So in this case

$\displaystyle 3\cdot 4^2 + 3^2\cdot 4 + 3^3\cdot{1\over 1-\frac34}$

Now what does this expression equal?

Get it? You probably shouldn't wait until the morning before to figure this stuff out :/