Helping a little, old (rich) lady

One day, on the drive home from work, I witnessed an elderly lady in battered clothes struggling. Her wheelie basket, piled with laundry, had gotten stuck. In an unexpected and unprecendented pang of empathy, I stopped and offered assistance. Delighted to have me after, apparently, struggling for some time I was quickly able to pop the stuck wheel out of its prison (the sewer grate).

Preparing to leave, she stops me. “Before you leave, I have a reward to offer!”

She has my attention now. “I can give you 1,000,000 dollars or I can give you thirty daily payments starting today of one penny. I shall double each subsequent payment.”

To be kind, I offer the “lower” payment of a penny. I don’t need any old ladies millions of dollars. She gives me a penny and departs.

Sitting back in my office, I stop and think about what I’ve done. I realize that each day she pays me (for all thirty of them) I can expect $2^{day-1}$ cents. Considering it further, I begin to understand that the miracle of a doubling scheme such as this is that every subsequent day I will receive exactly one more cent than all the previous days combined.

A few weeks later, it now takes a rather large dump truck to deliver the little copper coins. Wondering how many dollars this all amounts to I quickly scribble on a pad of paper: $\frac{2^{30}}{100} = \10,737,418$.

Just something to remember when you see an elderly lady in need.