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What Are Geometric Sequences?

Learn what a geometric sequence is and how it can be used to model everything from the balance of your bank account to the growth of populations.

By
Jason Marshall, PhD,
Episode #015

The Difference Between Geometric Growth and Arithmetic Growth

An investment like this pays what is called compound interest. It’s great for the investor 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 multiplicative factor.

[[AdMiddle]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 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

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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!

Math image courtesy of Shutterstock

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