25 Geometric progressions

The rate at which quantities grow when they are continuously doubling is fairly well recognized.

Although, it’s not something we always intuitively understand.

Transcript

In gambling, the method of doubling a bet every time you lose until a win is obtained (and hence returning the value of the original bet) is referred to as a martingale betting system, and is all very well and good if you can back your bet, but the problem is that runs of losses are not all that uncommon, and so starting with a $10 bet, losing 5 times in a row requires you to risk $320 on your next bet.

Population growth

If a population grows by 10% of its current size each month, how long will it take to double its size? If a population shrinks by 10% of its current size each month, how long will it take to halve its size?

 

Such progressions are also commonly observed in population growth.  Human population growth is not usually that fast, but the growth of bacteria is often expressed in terms of how quickly they double.  For example, on a hot day, bacteria in food can double every 20 minutes, so that a small amount of 100 bacteria on a piece of food at 8am in the morning could grow to over 26 million by 2pm.  We also might hear about online content and data increasing in similar ways.

This is often referred to as "exponential" growth, and where there is a constant factor between each term in the sequence, we can refer to it as a geometric progression.  Here is a graph of an exponential curve.

Writing geometric sequences algebraically

It might help to have a quick way to figure out the total if we started with the number 2 and doubled it 31 times, without having to go all the way through the sequence.  Expressing the sequence algebraically can help with this, meaning we can quickly calculate the total no matter what number we start with or how many times it has been doubled.

Let’s use [latex]a[/latex] to refer to the first term again, as we have done all through this chapter.  Next, we need to consider the ratio.   We’ll call this [latex]r[/latex]. The first number needs to be multiplied by something, which is then multiplied by itself, and multiplied by itself again, and so on.

So for the first three terms in our sequence we would have something that looks like this

[latex]a, a \times r, a \times r \times r...[/latex] or [latex]a, ar, ar^2[/latex]

The n^th term is then given by [latex]t_n=r^{n-1}[/latex].  There’s that [latex]n-1[/latex] term again – we explain why we do this in the chapter on Arithmetic Progressions.

Let’s see this formula at work.  In this case it’s tripling the number 2.

Fractions

However we have some different sorts of patterns when the ratio [latex]r[/latex] is a fraction.  For example, if [latex]r=\frac{1}{2}[/latex] what will happen when we start from 1?

We’ll have [latex]1, ½, ¼, 1/8, 1/16, 1/32, 1/64,[/latex] and so.   Hence the values are getting smaller each time.  This wasn’t the case with arithmetic progressions, an arithmetic progression would just be going up with a smaller increase than usual, e.g., [latex]1,1.5, 2, 2.5, 3, 3.5...[/latex]

This is because the effect of multiplying by a number between 0 and 1 is to make a number smaller.  This behaviour for geometric progressions is sometimes thought of as modelling half-life or decay.  The following Scratch code can help you plot geometric progressions for certain values (depending on r, they could very quickly become too large!).  You can also see the code on the project page.

Building intuition with progressions

Graph at least five terms from each of the following progressions – you can use the Scratch program above, try an online graphing tool, or graph them yourself.    You can even use an excel spreadsheet. Then write a sentence summarising the behaviour of the graph.  Try to think of an example of what might make a graph that looks like this (eg, the temperature of a cake over time as it comes out of an oven). If you’re struggling to understand the progressions below, take a look at the Scratch tutorial.

Arithmetic progressions	Geometric progressions
[latex]a > 0, d > 0[/latex] [latex]a < 0, d > 0[/latex] [latex]a > 0, d < 0[/latex] [latex]a < 0, d < 0[/latex]	[latex]a > 0, r > 1[/latex] [latex]a > 0, 0 < r < 1[/latex] [latex]a > 0, r < -1[/latex] [latex]a > 0, -1 < r < 0[/latex] [latex]a < 0, r > 1[/latex] [latex]a < 0, 0 < r < 1[/latex]