Recently a rare copper penny from 1792 was auctioned off in Florida for $1,150,000. In addition to lamenting the fact that my ancestors didn’t squirrel one of these away for me, my next thought was GEOMETRIC SEQUENCES! Isn’t 2012 simply another year on this penny’s valuable march of perpetuity? The penny fetched such a steep winning bid because it is both rare and in uncirculated condition.
I brought it up to my class and jump-started the wonderment: “How much did it go up in value each year?”
I was blown away by the thoughtful remarks:
1) “Well, it probably didn’t go up by the same amount each year.”
2) “It depends on if it went up by the same amount or the same percent.”
3) “It probably didn’t increase beyond its face value for years. In 1793 the penny was probably still worth a penny. And in 1801 it was probably still worth a penny.”
4) “Why would someone spend so much?”
5) “It probably depends on how many were minted.”
6) “What makes things valuable?”
I got more questions than we could answer. I especially liked the last question. I kept our focus trained on “If the penny was worth $.01 in 1792, and in 2012 it was worth $1,150,000, what percent did it increase per year?” full-well knowing that we were assuming constant growth, which would not be the case. Fresh out of a unit on sequences and series, we explored.
I had no idea what kind of answer we’d get, and I let them know this. Let’s put our math to work.
Solving for the common ratio, it turns out the pretty penny increased in value by about 8.8% every year. It’s as though the penny were tied to the stock market’s historical average rate of return, before there was a stock market! We verified our algebra, recursively charting the penny’s 220 year existence. I stored the exact value in the calculator, cleared my screen, initialized the sequence to 0.01, multiplied the answer by the stored ratio, which I named R. Every time I pressed ENTER the value of the penny revealed itself with each passing year.
After a minute or two of thrilling tedium, the value of the coin reached exactly $1,115,000. Priceless.
Extensions? If the coin continues at this rate of appreciation, when will it hit $2 million?
To Kill A Mockingbird, Apple, and a US Postage Stamp
Given a known starting value and a current value, what other interesting objects can we examine to determine their average rate of growth? I threw together a Growth Rate Comparison Worksheet to compare Apple’s stock value from its IPO to roughly today, a first edition of To Kill a Mockingbird, and the Inverted Jenny U.S. Postage Stamp. The worksheet is bare bones but should provide a place for everyone to give their “two cents!”