Mary's parents decided to give her an allowance every Saturday. The first Saturday they gave her $.01. For each successive week, they double the amount given the previous week. If Mary saves her allowance, in how many weeks will she become a millionaire?

If you approach this using pennies, then $1 miilion = 100,000,000 pennies.

The amount of pennies received each week is P_i = 2^i, where i=0 for week 1, i=1 for week 2, etc. The sum of all pennies received in week i plus all previous weeks is:


P_{total} = \Sigma P_i = 2^{(i+1)} -1.

So for the first few weeks you get:


\begin{matrix}

Week\ i & P_t\\
------ & ----- \\
0 & 1 \\
1 & 3 \\
2 & 7 \\
3 & 15 \\
4 & 31 \\
\end{matrix}


etc.

To determine when she will be a millionaire you need to solve for i :


2^{(i+1)} -1 = 100,000,000


Do you know how to solve for i?