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.
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.
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, 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. 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).