# Number sequences in nature.

I'm getting ready to teach the second calculus course in the 4 course sequence at my school. One of the required topics is an introduction to number sequences. I want to motivate this section with some interesting examples from our physical universe. I plan on including the Fibonacci sequence:

\begin{align} F(n+2) = F(n+1)+F(n);\qquad F(0)=1, F(1)=1 \end{align}

This one shows up in many places, Greek architecture, various plant life, rabbits! etc. I was wondering if any of you had some other interesting sequences with physical significance that would be nice for a calc 2 class discussion. Thx

• There is no reason to think it comes up in Greek architecture. Galileo got the sequence $1,3,5,7,\dots$ from inclined plane experiments. May 12, 2014 at 16:07
• @andre I thought Galileo worked out the relationship between time of roll down to angle. The odd naturals showed up here?
– JEM
May 12, 2014 at 16:10
• Pascal's Triangle and the many identities associated with it would be interesting. For example the Fibonacci Numbers are encoded in it. Other than combinatorial properties, other identities for the value of Pascal's triangle at certain places can be used for fast computation May 12, 2014 at 16:12
• @JEM: He wrote that the distances travelled in equal intervals of time grew like (constant times) $1,3,5,\dots$. Fancy way of saying total distance travelled behaves like $kt^2$. May 12, 2014 at 16:15
• @ Andre ok understood thx. @ Chris pascal very nice. I'll review the properties of it.
– JEM
May 12, 2014 at 16:16

How about the expected number of record annual high temperatures recorded in a given city -- with relevance to the possibility of climate change. If the temperature observed in a year has no dependence on the temperature observed in previous years, then the expected number of record highs in n years is:

$$H_n = \sum_{k=1}^{n} \frac{1}{k},$$

the sequence of partial sums of the harmonic series. So now you bring up sequences and series.

The expected number of records in the first year is by definition $1$. Given independence, the probability that the second year is a record is $1/2$ -- so the expected number of records for $2$ years is $1 + 1/2$. The probability that the thrd year is a record must be $1/3$ (look at the possible orderings with equal likelihood), and the expected number of records is now $1+1/2+1/3$, etc.

• Thx RRL, I think I can work this in. :)
– JEM
May 18, 2014 at 18:08