TWO to the Power - Our Campaign

Two to the Power Math

We start with ONE Person on ONE DAY making a $10 commitment AND providing TWO Referrals

\begin{equation} 2^{0} \times $10 = $10\end{equation}

The TWO referrals receive an email directing them to our website. If the TWO new referrals each make a ten dollar commitment on the 2nd day AND they each provide TWO referrals, we will have a total of $30 committed.

\begin{equation} 2^{1} \times $10 + 2^{0} \times $10 = $30 \end{equation}

If each new referral makes a ten dollar commitment on the 3rd day AND they each provide TWO referrals, we will have a total of $70 committed.

\begin{equation} 2^{2} \times $10 + 2^{1} \times $10 + 2^{0} \times $10 = $70 \end{equation}

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YOU GET THE IDEA

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SO HOW MANY DAYS WILL IT TAKE TO GET TO OUR GOAL OF $29,000,000 COMMITTED TO THIS BRIDGE?

\begin{equation} 2^{0} \times $10 + 2^{1} \times $10 + 2^{2} \times $10 + 2^{3} \times $10 + ...+ 2^{?} \times $10 = $29,000,000 \end{equation}

If you want, you can keep writing down the powers of two, multiplying by 10 and adding the total to the previous total until you get to the goal amount. That is one way to see how many days it will take to reach the $29,000,000 goal.

OR

WE CAN DO THE MATH (FUN ALERT!)

We can say that we are looking for the number, we will call it 'n' where the sum of the powers of 2 (people) multiplied by 10 (dollars) = $29,000,000.

We can express the formula using mathematical notation as follows:

\begin{equation} \sum_{i=0}^{n} 2^{i} \times $10 = $29,000,000\end{equation}

DID YOU NOTICE THAT...

\begin{equation} 2^{0} + 2^{1} + 2^{2} = 7 \mbox{ which also happens to be } 2^3 - 1 \end{equation}

AND

\begin{equation} 2^{0} + 2^{1} + 2^{2} +2^{3}= 15 \mbox{ which also happens to be } 2^4 - 1 \end{equation}

AND

\begin{equation} 2^{0} + 2^{1} + 2^{2} +2^{3} + 2^{4}= 31 \mbox{ which also happens to be } 2^5 - 1 \end{equation}

SO YOU MAY SEE THAT

\begin{equation} \sum_{i=0}^{n} 2^{i} = 2^{n+1} - 1\end{equation}

SO WE CAN WRITE THE FORMULA ABOVE AS

\begin{equation} (2^{n+1} - 1) \times $10 = $29,000,000\end{equation}

LET'S SOLVE FOR n

Divide both sides by $10

\begin{equation} 2^{n+1} - 1 = 2,900,000\end{equation}

Add 1 to both sides

\begin{equation} 2^{n+1} = 2,900,001\end{equation}

Get the Log of both sides

\begin{equation} (n+1)\log({2}) = \log({2,900,001})\end{equation}

Divide both sides by Log(2)

\begin{equation} (n+1) = \frac{\log({2,900,001})}{\log({2})} \end{equation}

Subtract 1 from both sides

\begin{equation} n = \frac{\log({2,900,001})}{\log({2})} - 1\end{equation}

GO AHEAD AND USE A CALCULATOR. YOU WON'T BE CHEATING!

\begin{equation} n = 21.46 - 1\end{equation}

SO

\begin{equation} n = 20.46\end{equation}

WHICH MEANS IT WOULD TAKE ABOUT 20 AND A HALF DAYS TO REACH OUR GOAL OF 2,900,000 PEOPLE COMMITTED TO DONATE $10 EACH!