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# Geometric Sequences

## Sequences with a common ratio

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Geometric Sequences

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. So tell me how many aliens there would be after they split 10 times,” Henry challenged.

In this concept, you will learn to recognize, extend and graph geometric sequences.

### Geometric Series

There are several different types of sequences that follow patterns. An arithmetic sequence has a fixed sum or difference between each term.

Consider 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 as 1, 4, 16, 64.

Is this an arithmetic sequence? 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. Since there is no common difference, this is not an arithmetic sequence.

Looking at the sequence, you may notice that each term is multiplied by 4 to get the next term. The fifth term in the sequence can then be found by taking the 4th term and multiplying by 4. Therefore the fifth term would be 64×4=256\begin{align*}64 \times 4=256\end{align*}.

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.

Let’s look at another example.

What is the common ratio between each of the terms in the sequence?

5,10, 20, 40, 80

Use a table to find the common ratio between successive terms.

 Terms Common Ratio 5\begin{align*}5\end{align*} 10\begin{align*}10\end{align*} 105=2\begin{align*}\frac{10}{5}=2\end{align*} 20\begin{align*}20\end{align*} 2010=2\begin{align*}\frac{20}{10}=2\end{align*} 40\begin{align*}40\end{align*} 4020=2\begin{align*}\frac{40}{20}=2\end{align*} 80\begin{align*}80\end{align*} 8040=2\begin{align*}\frac{80}{40}=2\end{align*}

The common ratio for this sequence is 2.

This is a geometric sequence where you multiply by 2 to get the next term in the sequence.

Let’s look at another example.

Consider the following sequence: 8, 24, 72, 216, ... What is the next term in the sequence?

First, determine if the sequence is arithmetic or geometric.

If it is arithmetic, it will have a common difference between successive terms. Use a table to determine the differences.

 Terms Difference 8\begin{align*}8\end{align*} 24\begin{align*}24\end{align*} 24−8=16\begin{align*}24-8=16 \end{align*} 72\begin{align*}72\end{align*} 72−24=48\begin{align*}72-24=48\end{align*} 216\begin{align*}216\end{align*} 216−72=144\begin{align*}216-72=144 \end{align*}

There is no common difference so the sequence is not arithmetic. If it is geometric, it will have a common ratio between successive terms. Use a table to determine the ratios.

 Terms Common Ratio 8\begin{align*}8\end{align*} 24\begin{align*}24\end{align*} 248=3\begin{align*}\frac{24}{8}=3\end{align*} 72\begin{align*}72\end{align*} 7224=3\begin{align*}\frac{72}{24}=3\end{align*} 216\begin{align*}216\end{align*} 21672=3\begin{align*}\frac{216}{72}=3\end{align*}

The sequence is geometric with a common ratio of 3.

Next, find the next term.

Since the common ratio is 3, the fifth term in the sequence will be 216×3=648\begin{align*}216 \times 3=648\end{align*}.

The sequence will go 8,24,72,216,648\begin{align*}8,24,72,216,{\color{red}648}\ldots\end{align*}.

It can be useful to graph geometric sequences. To do this, you would create a table of values and then use a coordinate plane to plot the points.

Let’s look at an example.

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.

First, create a table showing the 15 year span. The amount of space doubles every year so the common ratio is 2. This means to get the next term in the sequence you multiply the previous term by 2.

 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

Next, graph the data from the table. Again, you will have discrete data.

### Examples

#### Example 1

Earlier, you were given a problem about the splitting aliens.

In the video game that Sasha and Henry play, an alien splits into two aliens who then split into two more aliens every 10 minutes. Henry wants to know how many aliens there will be after 10 times.

You know that the aliens split into two every 10 minutes. Therefore the pattern is 2,4,8,16,32,\begin{align*}2,4,8,16,32,\ldots\end{align*}.

Continue this pattern for 10 times (or 100 minutes).

2,4,8,16,32,64,128,256,512,1024\begin{align*}2,4,8,16,32,64,128,256,512,1024\end{align*}

After the alien splits 10 times, there will be 1024 aliens in the game.

#### Example 2

Find the common ratio in the sequence.

14,18,116\begin{align*}\frac{1}{4},\frac{1}{8},\frac{1}{16}\end{align*}

Use a table to find the common ratio between successive terms.

 Terms Common Ratio 14\begin{align*}\frac{1}{4}\end{align*} 18\begin{align*}\frac{1}{8}\end{align*} 18÷14=12\begin{align*}\frac{1}{8} \div \frac{1}{4}= \frac{1}{2}\end{align*} 116\begin{align*}\frac{1}{16}\end{align*} 116÷18=12\begin{align*}\frac{1}{16} \div \frac{1}{8}= \frac{1}{2}\end{align*}

The answer is 12\begin{align*}\frac{1}{2}\end{align*}.

The common ratio for this sequence is 12\begin{align*}\frac{1}{2}\end{align*}.

This is a geometric sequence where you multiply by 12\begin{align*}\frac{1}{2}\end{align*} to get the next term in the sequence.

#### Example 3

Find the common ratio for the sequence 2, 4, 8, 16.

Use a table to find the common ratio between successive terms.

 Terms Common Ratio 2\begin{align*}2\end{align*} 4\begin{align*}4\end{align*} 4÷2=2\begin{align*}4 \div 2 =2 \end{align*} 8\begin{align*}8\end{align*} 8÷4=2\begin{align*}8 \div 4 =2\end{align*} 16\begin{align*}16\end{align*} 16÷8=2\begin{align*}16 \div 8 =2 \end{align*}

The common ratio for this sequence is 2.

This is a geometric sequence where you multiply by 2 to get the next term in the sequence.

#### Example 4

Find the common ratio for the sequence 1, 7, 49, 343.

Use a table to find the common ratio between successive terms.

 Terms Common Ratio 1\begin{align*}1\end{align*} 7\begin{align*}7\end{align*} 7÷1=7\begin{align*}7 \div 1 =7 \end{align*} 49\begin{align*}49\end{align*} 49÷7=7\begin{align*}49 \div 7 =7 \end{align*} 243\begin{align*}243\end{align*} \begin{align*}343 \div 49 =7 \end{align*}

The common ratio for this sequence is 7.

This is a geometric sequence where you multiply by 7 to get the next term in the sequence.

#### Example 5

Find the common ratio for the sequence 400, 100, 25.

Use a table to find the common ratio between successive terms.

 Terms Common Ratio \begin{align*}400\end{align*} \begin{align*}100\end{align*} \begin{align*}\frac{100}{400}=\frac{1}{4}\end{align*} \begin{align*}25\end{align*} \begin{align*}\frac{25}{100}=\frac{1}{4}\end{align*}

The answer is \begin{align*}\frac{1}{4}\end{align*}.

The common ratio for this sequence is \begin{align*}\frac{1}{4}\end{align*}.

This is a geometric sequence where you multiply by \begin{align*}\frac{1}{4}\end{align*} to get the next term in the sequence.

### Review

Find the common ratio between each term.

1. \begin{align*}-4,20,-100,500,-2500\end{align*}

2. \begin{align*}60,15,\frac{15}{4},\frac{15}{16}\end{align*}

3. \begin{align*} \frac{1}{8},\frac{1}{4},\frac{1}{2},1,2\end{align*}

4. \begin{align*}3,6,12,24\end{align*}

5. \begin{align*}4,2,1,0.5,0.25\end{align*}

6. \begin{align*}12,24,48\end{align*}

7. \begin{align*}\frac{1}{2},1,2\end{align*}

8. \begin{align*}100,50,25,12.5\end{align*}

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.

9. \begin{align*}1,4,7,10,13\end{align*}

10. \begin{align*}180,60,20,6 \frac{2}{3}\end{align*}

11. \begin{align*}102,94,86,78\end{align*}

12. \begin{align*}19,27,35,43,50\end{align*}

13. \begin{align*}5,-50,500,-5000,50000\end{align*}

14. \begin{align*} \frac{1}{6},\frac{1}{12},\frac{1}{24},\frac{1}{48}\end{align*}

15. \begin{align*}99,33,11\end{align*}

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### Vocabulary Language: English

geometric sequence

A geometric sequence is a sequence with a constant ratio between successive terms. Geometric sequences are also known as geometric progressions.

sequence

A sequence is an ordered list of numbers or objects.