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In this part of the course I am just trying to show that we actually see a lot of sequences and series every day in our regular life. I already found some examples such as the house numbers when you drive down a street, the number of people you reach in those 'chain mails', the value of your block in the game 2048, ...
I am still looking for some fun, everyday sequences or series that without much context give an idea of what those sequences are.
Here are a few more examples:
the amount on your savings account ;
the amount of money in your piggy bank if you deposit the same amount each week (a bank account with regular deposits leads you to arithmetico-geometric sequences) ;
the size of a population in exponential growth, e.g. bacteria in a Petri dish (or in your leftovers if you find Petri dishes not "every day life" enough) ;
the intensity of radioactivity after $n$ years of a given radioactive material (with application to determining the age of mommies!).
I tutored a student who came with a kind of problem I had never seen before and found quite refreshing. It was something like:
A child is being pushed on a swing by their father, reaching a maximum height of 4 feet. The father stops pushing, and the maximum height of the swing decreases by 15% on each successive swing.
I don't remember the question itself, but the main idea was that the sequence of maximum heights of the child was a geometric sequence! (Perhaps the goal was to find the total vertical distance traveled by the child.)
You get a related example considering a bouncing ball:
Image taken from Wikipedia
It turns out that a bouncing object loses an approximately constant fraction of its remaining energy with each bounce, and in turn the sequence of maximum heights is (approximately) geometric!
More info on the latter here.
I like to explain why arithmetic and geometric progressions are so ubiquitous. Using the examples other people have given.
Geometric progressions happen whenever each agent of a system acts independently. For example population growth each couple do not decide to have another kid based on current population. So population growth each year is geometric. Each radioactive atom independently disintegrates, which means it will have fixed decay rate. In other words that is why there is "half-life" of a radioactive element, in a fixed amount of time it becomes half. Email chains, Interest rate, etc are more examples of the same kind.
On the other end global/singular decisions give arithmetic progressions. If you add a fixed amount to your piggy bank each week that is arithmetic progression. The child who swings extra each time is likely to give only a constant extra force each time, so it is not likely for that to be geometric, it will be an arithmetic progression. There are exceptions of course like the ball bouncing is geometric even though it is singular because of coefficient of restitution. In general singular decisions can be anything - but typically arithmetic.
In reality, these are ideal cases, most of the natural phenomenon will have both global and local influencers. Making it somewhere in between arithmetic and geometric progressions.
If the population is already huge having another kid might not be so conducive. So the population growth will stop when overall resources get limited. Thomas Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources.
Tumour growth, the growth rate is exponential unless it becomes so large that it cannot get food to grow effectively. So it starts of exponentially and stops completely. A more precise statement is known as Gompertz Law of Mortality - "rate of decay falls exponentially with current size".
Even radioactive decay is not really immune, there is something called Quantum anti-Zeno effect if you wanna go wiki hopping.
This is a nice demonstration that $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\cdots = 2 \;.$$
They might be interested to know about both Moore's Law and "Nielsen's Law". You've probably heard about Moore's Law, where computer complexity doubles about every two and a half years.
Internet bandwidth seems to also have a doubling time, in this case that doubling time is 21 months.
When I think of a geometric sequence, I think of something where the initial input value = 1, not 0. Most interest problems would start at time = 0, so I would exclude these unless you said something like "let x = $ in bank at beginning of each year". Then your first input value would be 1.
Also, geometric sequences have a domain of only natural numbers (1,2,3,...), and a graph of them would be only points and not a continuous curved line. So again, a problem about earned interest might not be a perfect example, since you can withdraw your money at any instant and not only at whole number year values.
The best one I have come up with is tile values in the game 2048.
and so on...
Arithmetics Examples
Time on clock, each minute hand that the second hand covers is 5 seconds.
5, 10, 15, 20, 25, 30, ..., 60
The point at which a runner passes the finish line in a 3000 metre race.
200m, 600m, 1000m, ..., 2600m, 3000m
Sound waves or waves in the sea are sinusoids, so they can repeat their pattern for the range of the sinusoid.
Geometric examples
Half life of carbon or any element. You can then show how all the carbon 14 is depleted over thousands of years.
A graph where logs is used is easy to read and can be almost linear, whereas if there is a geometric increase you can't even plot it on paper. Kids love this one, and understand it very quickly.
Puzzle: If a frog is 1 metre from a door and jumps halfway, and then jumps halfway again continuing to half its jump each time, will it ever reach the door? This displays limits and geometric decreasing functions and the idea that decreasing jumps results in infinitesimally small results over time.
Showing kids how much they pay on a mortgage at 5% interest rate for 10 years, 20 years and 30 years is very insightful and clearly displays geometric increase.
