Theorem to prove (induction seems to be problematic)

${\displaystyle 3^{N}~=~\sum _{i=0}^{N}2^{i}{\binom {N}{i}}}$

where ${\displaystyle {\tbinom {N}{i}}={\tfrac {N!}{i!(N-i)!}}}$.

Induction gives

${\displaystyle 3\cdot 3^{N}~\sim ~2^{N+1}+\sum _{i=0}^{N}{\frac {N+1}{N+1-i}}\cdot 2^{i}{\binom {N}{i}}}$

Suggestions?

${\displaystyle \iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \mathbf {D} \;\cdot \mathrm {d} \mathbf {A} }$

${\displaystyle \int \!\!\!\!\!\int _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\bigcirc \,\,\mathbf {D} \;\cdot \mathrm {d} \mathbf {A} }$

The T.M.I. System: Randy Marsh tells the students that they can get their T.M.I. rating with the following formula:

"Length times diameter plus weight over girth divided by angle of the tip squared."

${\displaystyle {L\times d+{W \over G} \over \angle {\alpha }_{t}^{2}}\,\!}$

However according to the U.S. Surgeon General the correct formula for T.M.I. is

"Length times girth over angle of the shaft divided by mass over width."

${\displaystyle {{L\times G \over \angle {\alpha }_{s}}\div {M \over W}}}$

Cartman's psychiatrist claims "I have found a direct correlation between anger and T.M.I." and presents the equation

"Anger equals length times width over mass-squared minus angle of the shaft plus yaw"

${\displaystyle \mathrm {Anger} ={\frac {L\times W}{M^{2}-\angle {\alpha }_{s}}}+\mathrm {yaw} \,\!}$

However, the Surgeon General claims their official formula is "true solid science" and "100% accurate". To resolve the anger crisis, the Surgeon General goes on television to say the national average has lowered from 6 inches to 1.5 inches, thus anyone with a T.M.I. larger than 1.5 inches is considered "above average".

Note that T.M.I. is given in inches, a unit of length. Of the formulas given, only the Surgeon General's has consistent dimensions, but these are of length³ / mass. However, by redefining the measurement of some variables (such as defining "width" as a linear mass density), length alone can be produced. The scale of T.M.I. is similar to that of penis size.

Cartman gives the "Chemical equation for gold," as referenced in alchemy:

${\displaystyle gcfs_{2}-puc_{2}j_{2}+{\frac {cb(sn)}{d_{1}{\mathcal {O}}_{p}}}=GOLD}$

where gcfs is "guys with cash-for-gold signs", pucj is "people's unwanted crappy jewelry", cbsn is "cable-based shopping network", and d O_p is "demented old people".