Random Ramblings

With the next issue of the Bulletin, the 2012 Joint Mathematics Meeting, and SUMiT 2012 on the horizon, I haven’t had much time to blog.

But I wanted to mention a few things that caught my attention these last couple weeks.

First, congratulations to the members of Girls’ Angle for persevering so valiantly at the latest end-of-session math treasure hunt at Girls’ Angle last week. I love that several of the solutions the girls came up with were different from the ones I had in mind. I’ll write more about this in the next Bulletin.

Next, there’s Math or Mess’ latest post about telescoping sums. I just wanted to point out that telescoping sums can be used to find formulas for things like the sum of the first $N$ squares. Often such formulas are used as exercises in mathematical induction. But these leave the student wondering, how were the formulas deduced in the first place?

To use telescoping sums to find a nice expression, say, for the sum of the first $N$ squares, one observes that $(k + 1)^3 - k^3 = 3k^2 + 3k + 1$. If one sums over $k$ from 1 to $N$, the left hand side telescopes and we get:

$(N+1)^3 - 1^3 = 3S_N + 3(\frac{N(N+1)}{2}) + N$,

where $S_N$ is the sum of the first $N$ squares. One can then simplify to obtain the well-known formula $S_N = \frac{N(N+1)(2N+1)}{6}$. (Another way to deduce the formula is to observe that because $S_{N+1} - S_N$ is a quadratic, the formula for $S_N$ must be a cubic in $N$. One can then determine which cubic passes through the first 4 values of $S_N$.)

And then, I love this line from Mathspig’s Zombie’s series “Another way of calculating a zombie’s velocity or speed is to use direct proportion.” Somehow, it really captures the essence of zombie movement! I don’t know if this makes me frightened of direct proportions now or if this just makes zombies seem simple and tame…but direct proportions are one of those basic math concepts that pervade all kinds of phenomena, such as brownie cookie time (which is proportional to brownie thickness), earnings (if you’re paid on an hourly rate or something similar)…even Einstein’s famous formula $E = mc^2$ is an example of direct proportion (assuming that the speed of light in a vacuum is, in fact, a constant).

Finally, there’s a new article by Jonathan Kane and Janet Mertz in the Notices of the American Mathematical Society: Debunking Myths About Gender and Mathematics Performance. I confess to not having had a chance to read it yet, but I still wanted to mention it because the contents seem interesting and several people have pointed it out to me.