12.19: Geometric Sequences
Have you ever been to an arcade? Take a look at this dilemma.
On the way home from school on the day of the trip downtown, a bunch of students stopped off at the arcade. It was always fun to talk and get something to eat and play a video game or two. Sam and Henry began to play a favorite game of theirs with aliens.
“That has a lot of math in it,” Sasha commented as Henry had his turn.
“How do you figure?” Henry asked.
“It just does,” Sasha said convincingly. “Think about it. In this video game, an alien splits into two aliens who then split into two more aliens every 10 minutes.”
“Good point, how many aliens there would be after they split 10 times?” Henry asked.
We can start by thinking about this as a number pattern. This Concept is all about patterns and sequences. Think about the video game and you will need to solve the sequence at the end of the Concept.
Guidance
Have you ever seen a number sequence? There are several different types of sequences that follow patterns.
An arithmetic sequence has a fixed sum or difference between each term.
Now look at the boxes below, and you will see another type of sequence.
Can you see a pattern? The boxes increase each time. Using numbers, the sequence could be written 1, 4, 16, 64. You might even guess at what would come next. Is there a common difference between them? Not really. There is a difference of 3 between the first two terms, 12 between the second and third terms, and 48 between the third and fourth terms. If you guessed that 256 would follow it’s because you figured out the pattern. You noticed that to get to the next term, you have to multiply by 4 instead of by adding a certain number.
This is a geometric sequence; it’s a sequence in which the terms are found by multiplying by a fixed number called the common ratio. In the situation above, the common ratio is 4.
Once you know the common ratio, then you can figure out the next step in the pattern.
Take a look at this one.
What is the common ratio between each of the terms in the sequence?
The ratio is 2 between each number.
You can see how knowing the common ratio helped us with our problem solving.
Consider the following sequence:
\begin{align*} 8, 24, 72, 216, \ldots\end{align*}
Doesn’t your brain want to find the next number? You’ve probably figured out that the common ratio here is 3. So the next term in the sequence would be \begin{align*}216 \cdot 3\end{align*} or 648. You would continue the same process to find the term that follows. Or, you could divide by 3 to find the previous term.
Just as we did with arithmetic sequences, it can be useful to graph geometric sequences. We’ll use the same method as before—create a table of values and then use a coordinate plane to plot the points.
Take a look at this situation.
The amount of memory that computer chips can hold in the same amount of space doubles every year. In 1992, they could hold 1MB. Chart the next 15 years in a table of values and show the amount of memory on the same size chip in 2007.
Year | Memory (MB) |
---|---|
1992 | 1 |
1993 | 2 |
1994 | 4 |
1995 | 8 |
1996 | 16 |
1997 | 32 |
1998 | 64 |
1999 | 128 |
2000 | 256 |
2001 | 512 |
2002 | 1024 |
2003 | 2048 |
2004 | 4096 |
2005 | 8192 |
2006 | 16384 |
2007 | 32768 |
Find the common ratio for each sequence.
Example A
\begin{align*}2, 4, 8, 16\end{align*}
Solution: \begin{align*}2\end{align*} is the common ratio.
Example B
\begin{align*}1, 7, 49, 343\end{align*}
Solution: \begin{align*}7\end{align*} is the common ratio.
Example C
\begin{align*}400, 100, 25\end{align*}
Solution: \begin{align*}\frac{1}{4}\end{align*} is the common ratio.
Now let's go back to the dilemma from the beginning of the Concept.
We can write a number pattern.
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
1024 aliens after 10 splits!
This is the answer to our problem.
Vocabulary
- Sequence
- a series of numbers that follows a pattern.
- Geometric Sequence
- a sequence where you find terms by multiplying a fixed number by a common ratio.
Guided Practice
Here is one for you to try on your own.
Find the common ratio in the sequence.
\begin{align*}\frac{1}{4}, \frac{1}{8}, \frac{1}{16}\end{align*}
Solution
Each of the values was divided in half. Therefore, the common ratio is \begin{align*}\frac{1}{2}\end{align*}.
This is the answer.
Video Review
Practice
Directions: Find the common ratio between each term.
- \begin{align*}-4, 20, -100, 500, -2500\end{align*}
- \begin{align*}60, 15,\end{align*} \begin{align*} \frac{15}{4}\end{align*}, \begin{align*} \frac{15}{16}\end{align*}
- \begin{align*}\frac{1}{8}\end{align*}, \begin{align*}\frac{1}{4}\end{align*}, \begin{align*}\frac{1}{2}\end{align*}, 1, 2
- \begin{align*}3, 6, 12, 18\end{align*}
- \begin{align*}4, 2, 1, .5, .25\end{align*}
- \begin{align*}12, 24, 48\end{align*}
- \begin{align*}\frac{1}{2}, 1, 2\end{align*}
- \begin{align*}100, 50, 25, 12.5\end{align*}
Directions: Identify the following sequences as an arithmetic sequence, a geometric sequence, or neither. For arithmetic sequences, find the common difference. For geometric sequences, find the common ratio.
- 1, 4, 7, 10, 13
- 180, 60, 20, \begin{align*}6 \frac{2}{3}\end{align*}
- 102, 94, 86, 78
- 18, 27, 35, 43, 50
- 5, -50, 500, -5000, 50000
- \begin{align*} \frac{1}{6}\end{align*}, \begin{align*} \frac{1}{12}\end{align*}, \begin{align*} \frac{1}{24}\end{align*}, \begin{align*} \frac{1}{48}\end{align*}
- \begin{align*}99, 33, 11\end{align*}
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Image Attributions
Here you'll recognize, extend and graph geometric sequences.