![]() ![]() 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. 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. Kids love this one, and understand it very quickly. You can then show how all the carbon 14 is depleted over thousands of years.Ī 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. Determining the basic graph of each type of sequence Writing explicit formulas for geometric and arithmetic sequences. Questions involve: Finding the nth term of geometric and arithmetic sequences. Sound waves or waves in the sea are sinusoids, so they can repeat their pattern for the range of the sinusoid. This is a set of 16 task cards involving explicit formulas for geometric and arithmetic sequences. The point at which a runner passes the finish line in a 3000 metre race. Time on clock, each minute hand that the second hand covers is 5 seconds. A more precise statement is known as Gompertz Law of Mortality - "rate of decay falls exponentially with current size".Įven radioactive decay is not really immune, there is something called Quantum anti-Zeno effect if you wanna go wiki hopping. So it starts of exponentially and stops completely. Tumour growth, the growth rate is exponential unless it becomes so large that it cannot get food to grow effectively. 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. So the population growth will stop when overall resources get limited. If the population is already huge having another kid might not be so conducive. Making it somewhere in between arithmetic and geometric progressions. In reality, these are ideal cases, most of the natural phenomenon will have both global and local influencers. In general singular decisions can be anything - but typically arithmetic. There are exceptions of course like the ball bouncing is geometric even though it is singular because of coefficient of restitution. 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. If you add a fixed amount to your piggy bank each week that is arithmetic progression. On the other end global/singular decisions give arithmetic progressions. Email chains, Interest rate, etc are more examples of the same kind. In other words that is why there is "half-life" of a radioactive element, in a fixed amount of time it becomes half. Each radioactive atom independently disintegrates, which means it will have fixed decay rate. So population growth each year is geometric. For example population growth each couple do not decide to have another kid based on current population. Geometric progressions happen whenever each agent of a system acts independently. Using the examples other people have given. I like to explain why arithmetic and geometric progressions are so ubiquitous. ![]()
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