Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.
Here’s a question I once received:
If the pattern goes on, and if , what is ?
I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.
Let’s calculate the first few terms to try to find a pattern:
Written another way,
Continuing the pattern, we see that , or .
If you try plugging that number into your calculator, you’ll probably get an error. Fortuniately, we can use logarithms to approximate the answer. Since , we have
Plugging into a calculator, we find that
When this actually happened to me, it took me about 10 seconds to answer — without a calculator — “I’m not sure, but I do know that the answer has about 10 million digits.” Naturally, my class was amazed. How did I do this so quickly? I saw that the answer was going to be , so I used the approximation to estimate
Next, I had memorized the fact that that . So I multiplied by to get approximately 10 million. As it turned out, this approximation was a lot more accurate than I had any right to expect.