数学英语 18 What Are Geometric Sequences?

by Jason Marshall

In the last article, we began talking about mathematical sequences—in particular, a type of sequence known as an arithmetic sequence. Today, we’re going to turn our attention to another type of sequence called a geometric sequence. In addition to talking about the math, we’ll also cover how these sequences can be used to model many aspects of the natural world—including the balance of your bank account and the growth of populations.

But first, the podcast edition of this article was sponsored by Go To Meeting. Save time and money by hosting your meetings online. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their web conferencing solution.

Review of Arithmetic Sequences

Let’s start by quickly reviewing what we talked about last time. Sequences in math are simply lists of numbers arranged in a particular order: 1, -5, 3, 10 is one example of an infinite number of possible sequences. In the last article, we specifically looked at a special type of sequence called an arithmetic sequence. In that type of sequence, the difference between any two successive elements is always the same constant value.

For example, 3, 6, 9, 12, 15 is an arithmetic sequence since the difference between each successive element is 3. If we were to add one more element to the end of the sequence, what would it be? Well, the last element currently is 15, and the difference between successive elements is 3, so the next element in the sequence 3, 6, 9, 12, 15 would have to be 15 + 3 = 18.

What is a Geometric Sequence?

Okay, let’s move on to another special type of sequence called a geometric sequence. Whereas sequential elements in arithmetic sequences differ by a constant offset 1, sequential elements in geometric sequences differ by a constant ratio. The easiest way to explain what I mean is with an example. Consider the sequence 1, 2, 4, 8, 16. That is a geometric sequence because each successive element is obtained by multiplying the previous one by 2. So, what’s the next element in the sequence? Well, the current last element is 16, so 2 x 16 = 32.

Geometric Sequences and Your Bank Account

So where do geometric sequences show up in your life? Well, the first place you might want to look is your bank account. If you invest money in a compound-interest-earning account, then your initial investment will grow as a geometric sequence. Say you invest $1000 in an account that pays 5% interest compounded annually 2. That means that every year the value of your account will grow by 5%.

For the first year, 5% of $1000 is $50 (remember, you can easily find this by noting that 10% of $1000 is $100, so 5% of $1000 must therefore be half of this). So after one year you have $1000 + $50 = $1050 in your account. Now, for the second year, 5% of $1050 is $52.50, so your balance after two years will be $1050.00 + $52.50 = $1102.50. Continuing on, after three years you have a total of about $1158, and after 4 years you have about $1216. Now, let’s take a look at this sequence (rounded to the nearest dollar): $1000, $1050, $1103, $1158, $1216. If we divide any two successive elements in this sequence—e.g., 1216 / 1158 or 1103 / 1050—the ratio is always 1.05 (which, by the way, is just 1 plus the interest rate written as a decimal).

The Difference Between Geometric Growth and Arithmetic Growth

An investment like this pays what is called compound interest. It’s great for the investor 3 because all earned interest begins to earn even more interest on itself. This sort of growth—the type that results in a geometric sequence of numbers—is called, logically, geometric growth. In other words, when something grows geometrically, its value always increases by a fixed 4 multiplicative factor.

Now, if the interest in our example was not compounded, then the investment would have simply continued to earn $50 per year—5% of the initial investment—forever. That type of growth is called arithmetic growth, since the total value of the investment each year is simply $50 greater—a constant value—than it was the previous year. So, after the same four years, the investment would be worth only $1200 instead of the $1216 with compound interest. Now, that isn’t a huge amount, but if the initial investment had been larger, or it had been allowed to grow for many years—perhaps decades—the difference could be huge. In fact, it would take about 14 years for the initial investment to double with geometric growth, but 20 years to double with arithmetic growth!

Geometric Sequences and Population Growth

But geometric sequences and geometric growth don’t only apply to the financial world. The growth of populations of living creatures can also be thought of in terms of a geometric sequence. In the simplest case, if two organisms reproduce enough times to replace themselves in the first reproductive cycle, then the total number of organisms becomes four. Similarly, after the second cycle, the total doubles to eight—it doubles each time. So, this sequence—2, 4, 8, 16, 32, …—is a geometric sequence. And it gets very large, very fast. By the twentieth cycle, there would already be more than one million organisms!

But is this how nature really works? Is this too simplistic? Well, perhaps…

Brain-Teaser Problem

Next time, we’ll continue our tour of mathematical sequences with a look at the famous Fibonacci sequence! Until then, here’s a question dealing 5 with geometric sequences for you to think about:

Is there a problem with modeling the growth of populations as a geometric sequence?

Hint: This might have something to do with the Fibonacci sequence. Think about it, and then be sure to look for my explanation in this week’s Math Dude Video Extra! episode on YouTube and Facebook.

Wrap Up

That’s all the math we have time for today. Thanks again to our sponsor this week, Go To Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.

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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!