Geometric sequences have several real-world applications and can be used to model situations ranging from physics to finance. One physical example would be if you had a rubber bouncy ball and dropped it from a height of 10 feet off the ground and measured the ball's height after each subsequent bounce. You'll find the list of heights approximately follows a geometric sequence. In finance, the total amount of money in a savings account earning interest over time follows a geometric sequence (assuming no deposits or withdrawals).
In order to fully understand and use geometric sequences, it might be worth brushing up on a few topics such as exponents and sequences. Much like an arithmetic sequence, a geometric sequence is an ordered list of numbers with a first term, second term, third term, and so on.
Any given geometric sequence is defined by two parameters: its initial term and its common ratio. The initial term is the name given to the first number on the list and the common ratio is the amount that each successive number in the list is multiplied by to get the next number. We denote the common ratio as and the initial term as . Subsequent terms are named similarly:
We will refer to a general term or nth term in this sequence as , and the next term would therefore be . To get from to in a geometric sequence, we multiply by the common ratio .
This gives us the rule for any two successive terms:
We can now use this rule to derive a formula for any term we want!
We start by finding from a given initial term .
Since we know what is in terms of we can substitute that back in to also get also in terms of :
As we continue to back substitute to get each term in the sequence using only the initial term and the common ratio, we can see that a pattern is starting to form. Writing this pattern down formally gives us an explicit formula for finding any term we want without having to write down all the terms that came before it:
So, the 10th term would be the initial term times the common ratio to the 9th power.
One graphical way of representing a geometric sequence would be by making a single rectangle with the height . Then, each new rectangle to its right would be some common ratio, say 80%, of the height of the rectangle to its left. In this example, we can see that the rectangles get smaller; had the common ratio been larger than one, they would have gotten taller.
Let's look at a few concrete examples to get a feel for these kinds of sequences.
Case 1: The common ratio is greater than one:
Case 2: The common ratio is between zero and one:
Case 3: The common ratio is equal to one:
Takeaways: From these examples we can see that the common ratio, r , determines the behavior of the sequence. If the value for r is greater than one, the list of terms grows larger without bound as in Case 1. If r is less than one, the terms get smaller and smaller approaching zero as in Case 2. If r is equal to one, the sequence remains stagnant as in Case 3.
1. Given and , list the first 5 terms of the geometric sequence.
Using the explicit formula for the geometric sequence, we can generate the list of terms:
Substitute in the values for the initial term, common ratio, and n:
Plug these values into a calculator or do the multiplication by hand to get the first 5 terms of the sequence as desired:
2. Given and , what is the term of the geometric sequence?
3. If the term is 4 and the term is 32, what are the common ratio and initial term?
Unlike the previous question where we were provided both the common ratio and the initial term, we are instead given two of its terms.
Let's start by writing out as much of the sequence as we know:
?, 4, ?, ?, 32, …
We can then also write down the formulas for the terms we know:
We now have a system of two equations with two unknowns, and if we divide equation (II) by equation (I) we will eliminate the variable :
Since we now know the common ratio is 2, we can solve for the initial term using equation (I) being more explicit:
So, the common ratio and the initial term are both equal to 2 in this case, and the sequence is the list of powers of two: 2, 4, 8, 16, 32, 64, …
Q: Can we think about a geometric sequence in terms of division?
A: Yes! Since multiplication and division are inverse operations of each other, if multiplication by the common ratio gets you the number to the right, then division by the common ratio gets you the previous number to the left, though you need to be careful not to go past the first term.
Q: How do geometric sequences and series differ?
A: In a geometric series, we are interested not in just listing the terms in order, but in finding the sum total value of those terms, so in a series, we might write: In a geometric sequence, we are instead interested in up to some total number of terms.
Q: What is an infinite geometric sequence?
A: An infinite geometric sequence is one that does not stop after a set number of terms, and depending on the common ratio and initial value, that might either tend towards zero or infinity. An infinite geometric series is one where an infinite list of terms is added together.
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