MTH120 College Algebra Chapter 9.3

 9.3 Geometric Sequences

Geometric sequences are a type of sequence in which each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (often denoted as "r"). The general formula for a geometric sequence is:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

Where:

• ${�}_{�}$ is the nth term of the sequence.
• ${�}_1$ is the first term of the sequence.
• $�$ is the common ratio.
• $�$ is the position of the term.

Geometric sequences are often used to model exponential growth or decay. They are prevalent in various fields, including finance, biology, physics, and computer science.

Here are some key points about geometric sequences:

1. Common Ratio (r): In a geometric sequence, the common ratio represents how much each term is multiplied by to obtain the next term. If $�>1$, the sequence exhibits exponential growth, while if $0<�<1$, the sequence shows exponential decay.

2. First Term (${�}_1$): The first term in the sequence is denoted as ${�}_1$. It serves as the starting point for the sequence.

3. General Formula: The general formula for a geometric sequence is handy for finding any term in the sequence directly without having to compute each preceding term.

4. Sum of a Geometric Series: You can find the sum of a finite geometric series using the formula:

   ${�}_{�}=\frac{{�}_1\cdot (1-{�}^{�})}{1-�}$

   Where ${�}_{�}$ is the sum of the first n terms of the sequence.

Geometric sequences are a fundamental concept in mathematics and have many practical applications, such as compound interest in finance, population growth in biology, and exponential decay in physics. Understanding geometric sequences is essential for various areas of science and engineering.

To find the common ratio (r) in a geometric sequence, you can use the following formula:

$�=\frac{{�}_{�+1}}{{�}_{�}}$

Where:

• $�$ is the common ratio.
• ${�}_{�}$ is the nth term of the sequence.
• ${�}_{�+1}$ is the term that follows the nth term in the sequence.

Let's go through a couple of examples to find the common ratio in geometric sequences:

Example 1: Finding the Common Ratio in a Geometric Sequence

Consider the geometric sequence: 2, 6, 18, 54, ...

To find the common ratio, you can choose any two consecutive terms in the sequence. Let's find the common ratio between the first and second terms and then check if it's the same for the other terms.

• ${�}_1=2$
• ${�}_2=6$

Now, use the formula for the common ratio:

$�=\frac{{�}_2}{{�}_1}=\frac{6}{2}=3$

The common ratio is 3.

To confirm that the common ratio is the same for the other terms, let's check the next pair of terms:

• ${�}_2=6$
• ${�}_3=18$

Again, using the formula:

$�=\frac{{�}_3}{{�}_2}=\frac{18}{6}=3$

The common ratio remains 3 for all pairs of consecutive terms. Therefore, the common ratio for this sequence is indeed 3.

Example 2: Finding the Common Ratio in a Geometric Sequence

Consider the geometric sequence: 8, 4, 2, 1, ...

To find the common ratio, choose any two consecutive terms in the sequence:

• ${�}_1=8$
• ${�}_2=4$

Use the formula for the common ratio:

$�=\frac{{�}_2}{{�}_1}=\frac{4}{8}=0.5$

The common ratio is 0.5.

You can verify that the common ratio is consistent for all pairs of consecutive terms in the sequence:

• ${�}_2=4$, ${�}_3=2$: $�=\frac{2}{4}=0.5$
• ${�}_3=2$, ${�}_4=1$: $�=\frac{1}{2}=0.5$

The common ratio is the same for all pairs of consecutive terms, which confirms that it's 0.5 for this sequence.

These examples demonstrate how to find the common ratio in geometric sequences. The common ratio is crucial for understanding the behavior and growth/decay patterns of geometric sequences.

A geometric sequence, also known as a geometric progression, is a sequence of numbers in which each term is found by multiplying the preceding term by a fixed, non-zero number called the "common ratio." In a geometric sequence, the ratio of any two consecutive terms is constant. The general form of a geometric sequence is as follows:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

Where:

• ${�}_{�}$ represents the nth term of the sequence.
• ${�}_1$ is the first term of the sequence.
• $�$ is the common ratio.
• $�$ is the position of the term.

Key features of a geometric sequence:

1. Common Ratio (r): The common ratio is the factor by which each term is multiplied to obtain the next term. It's the most defining characteristic of a geometric sequence. If $�>1$, the sequence exhibits exponential growth, while if $0<�<1$, the sequence demonstrates exponential decay.

2. First Term (${�}_1$): ${�}_1$ is the initial term or the starting point of the sequence.

3. General Formula: The general formula allows you to calculate any term in the sequence directly without having to compute all the preceding terms.

Geometric sequences are widely used to model various real-world situations involving exponential growth or decay, such as compound interest in finance, population growth in biology, and exponential decay in physics. Understanding geometric sequences is fundamental in mathematics and has applications in numerous fields of science and engineering.

To write the terms of a geometric sequence, you need to know the first term (${�}_1$) and the common ratio ($�$) of the sequence. The general formula for the nth term (${�}_{�}$) of a geometric sequence is:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

Here are examples of how to write the terms of geometric sequences given ${�}_1$ and $�$:

Example 1: Writing the Terms of a Geometric Sequence

Suppose you have a geometric sequence with the following information:

• First term (${�}_1$) is 2.
• Common ratio ($�$) is 3.

To find the first five terms of this sequence:

• For $�=1$: ${�}_1=2\cdot 3^{(1-1)}=2\cdot 3^0=2\cdot 1=2$

• For $�=2$: ${�}_2=2\cdot 3^{(2-1)}=2\cdot 3^1=2\cdot 3=6$

• For $�=3$: ${�}_3=2\cdot 3^{(3-1)}=2\cdot 3^2=2\cdot 9=18$

• For $�=4$: ${�}_4=2\cdot 3^{(4-1)}=2\cdot 3^3=2\cdot 27=54$

• For $�=5$: ${�}_5=2\cdot 3^{(5-1)}=2\cdot 3^4=2\cdot 81=162$

So, the first five terms of this geometric sequence are 2, 6, 18, 54, and 162.

Example 2: Writing the Terms of a Geometric Sequence

Suppose you have another geometric sequence with the following information:

• First term (${�}_1$) is 5.
• Common ratio ($�$) is 0.5.

To find the first five terms of this sequence:

• For $�=1$: ${�}_1=5\cdot 0.5^{(1-1)}=5\cdot 0.5^0=5\cdot 1=5$

• For $�=2$: ${�}_2=5\cdot 0.5^{(2-1)}=5\cdot 0.5^1=5\cdot 0.5=2.5$

• For $�=3$: ${�}_3=5\cdot 0.5^{(3-1)}=5\cdot 0.5^2=5\cdot 0.25=1.25$

• For $�=4$: ${�}_4=5\cdot 0.5^{(4-1)}=5\cdot 0.5^3=5\cdot 0.125=0.625$

• For $�=5$: ${�}_5=5\cdot 0.5^{(5-1)}=5\cdot 0.5^4=5\cdot 0.0625=0.3125$

So, the first five terms of this geometric sequence are 5, 2.5, 1.25, 0.625, and 0.3125.

These examples demonstrate how to calculate and write the terms of geometric sequences when you know the first term and the common ratio.

To find the first four terms of a geometric sequence when you know the first term (${�}_1$) and the common ratio ($�$), you can use the general formula for the nth term of a geometric sequence:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

Let's find the first four terms:

1. First Term: ${�}_1$
2. Second Term: ${�}_2={�}_1\cdot �$
3. Third Term: ${�}_3={�}_1\cdot {�}^2$
4. Fourth Term: ${�}_4={�}_1\cdot {�}^3$

So, the formula is:

1. First Term: ${�}_1$
2. Second Term: ${�}_1\cdot �$
3. Third Term: ${�}_1\cdot {�}^2$
4. Fourth Term: ${�}_1\cdot {�}^3$

Here's an example:

Suppose you have a geometric sequence with the first term (${�}_1$) as 3 and the common ratio ($�$) as 2. To find the first four terms of this sequence:

1. First Term: ${�}_1=3$
2. Second Term: ${�}_2=3\cdot 2=6$
3. Third Term: ${�}_3=3\cdot 2^2=3\cdot 4=12$
4. Fourth Term: ${�}_4=3\cdot 2^3=3\cdot 8=24$

So, the first four terms of this geometric sequence are 3, 6, 12, and 24, respectively.

Recursive formulas are a way to define a geometric sequence by specifying the first term (${�}_1$) and a formula for generating each subsequent term in terms of the previous term. In a geometric sequence, the recursive formula typically takes the form:

${�}_{�+1}={�}_{�}\cdot �$

Where:

• ${�}_{�+1}$ is the next term in the sequence (term n+1).
• ${�}_{�}$ is the current term in the sequence (term n).
• $�$ is the common ratio of the sequence.

Let's go through some examples of geometric sequences and their recursive formulas:

Example 1: Geometric Sequence with Recursive Formula

Suppose you have a geometric sequence with the following information:

• First term (${�}_1$) is 4.
• Common ratio ($�$) is 3.

You can use the recursive formula to generate the terms in this sequence:

• First Term: ${�}_1=4$
• Recursive Formula: ${�}_{�+1}={�}_{�}\cdot 3$

Now, let's generate the first few terms using the recursive formula:

• For $�=1$, you already know ${�}_1$, which is 4.
• For $�=2$: ${�}_2={�}_1\cdot 3=4\cdot 3=12$
• For $�=3$: ${�}_3={�}_2\cdot 3=12\cdot 3=36$
• For $�=4$: ${�}_4={�}_3\cdot 3=36\cdot 3=108$

The first four terms in this geometric sequence are 4, 12, 36, and 108.

Example 2: Geometric Sequence with Recursive Formula

Suppose you have another geometric sequence with the following information:

• First term (${�}_1$) is 10.
• Common ratio ($�$) is 0.5.

Using the recursive formula, you can generate the terms in this sequence:

• First Term: ${�}_1=10$
• Recursive Formula: ${�}_{�+1}={�}_{�}\cdot 0.5$

Now, let's generate the first few terms using the recursive formula:

• For $�=1$, you already know ${�}_1$, which is 10.
• For $�=2$: ${�}_2={�}_1\cdot 0.5=10\cdot 0.5=5$
• For $�=3$: ${�}_3={�}_2\cdot 0.5=5\cdot 0.5=2.5$
• For $�=4$: ${�}_4={�}_3\cdot 0.5=2.5\cdot 0.5=1.25$

The first four terms in this geometric sequence are 10, 5, 2.5, and 1.25.

These examples demonstrate how to use the recursive formula to generate the terms of a geometric sequence based on the first term and the common ratio.

Explicit formulas are a way to define a geometric sequence by specifying the first term (${�}_1$) and the common ratio ($�$), along with a formula that directly calculates any term in the sequence without relying on the previous term. In a geometric sequence, the explicit formula is:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

Where:

• ${�}_{�}$ is the nth term of the sequence.
• ${�}_1$ is the first term of the sequence.
• $�$ is the common ratio.
• $�$ is the position of the term.

Let's go through some examples of geometric sequences and their explicit formulas:

Example 1: Geometric Sequence with Explicit Formula

Suppose you have a geometric sequence with the following information:

• First term (${�}_1$) is 2.
• Common ratio ($�$) is 3.

The explicit formula for this geometric sequence is:

${�}_{�}=2\cdot 3^{(�-1)}$

Using this formula, you can calculate the terms directly. Let's find the first few terms:

• For $�=1$: ${�}_1=2\cdot 3^{(1-1)}=2\cdot 3^0=2\cdot 1=2$

• For $�=2$: ${�}_2=2\cdot 3^{(2-1)}=2\cdot 3^1=2\cdot 3=6$

• For $�=3$: ${�}_3=2\cdot 3^{(3-1)}=2\cdot 3^2=2\cdot 9=18$

• For $�=4$: ${�}_4=2\cdot 3^{(4-1)}=2\cdot 3^3=2\cdot 27=54$

The first four terms in this geometric sequence are 2, 6, 18, and 54.

Example 2: Geometric Sequence with Explicit Formula

Suppose you have another geometric sequence with the following information:

• First term (${�}_1$) is 8.
• Common ratio ($�$) is 0.5.

The explicit formula for this geometric sequence is:

${�}_{�}=8\cdot (0.5{)}^{(�-1)}$

Using this formula, you can calculate the terms directly. Let's find the first few terms:

• For $�=1$: ${�}_1=8\cdot (0.5{)}^{(1-1)}=8\cdot (0.5^0)=8\cdot 1=8$

• For $�=2$: ${�}_2=8\cdot (0.5{)}^{(2-1)}=8\cdot (0.5^1)=8\cdot 0.5=4$

• For $�=3$: ${�}_3=8\cdot (0.5{)}^{(3-1)}=8\cdot (0.5^2)=8\cdot 0.25=2$

• For $�=4$: ${�}_4=8\cdot (0.5{)}^{(4-1)}=8\cdot (0.5^3)=8\cdot 0.125=1$

The first four terms in this geometric sequence are 8, 4, 2, and 1.

These examples demonstrate how to use the explicit formula to directly calculate the terms of a geometric sequence based on the first term and the common ratio.

Geometric sequences are often used to solve application problems that involve exponential growth or decay. These problems can be found in various fields such as finance, population dynamics, physics, and engineering. To solve these problems, you'll typically need to identify the geometric sequence, find the common ratio and the first term, and then use the formula for the nth term of a geometric sequence.

Let's go through a couple of examples of solving application problems with geometric sequences:

Example 1: Compound Interest

Suppose you have $1,000 to invest in a savings account, and the account earns an annual interest rate of 5%. The interest is compounded annually. You want to find out how much money you will have in the account after 10 years.

To solve this problem, you can use the formula for compound interest:

$�=�{(1+\frac{�}{�})}^{��}$

Where:

• $�$ is the future amount (the amount you want to find).
• $�$ is the principal amount (initial deposit), which is $1,000.
• $�$ is the annual interest rate (in decimal form), which is 5% or 0.05.
• $�$ is the number of times interest is compounded per year, which is 1 for annually.
• $�$ is the number of years, which is 10.

Substitute the values into the formula:

$�=1000{(1+\frac{0.05}{1})}^{1\cdot 10}$

Simplify:

$�=1000\cdot (1.05{)}^{10}$

Now, you can see that this problem involves a geometric sequence with the first term (${�}_1$) being $1,000 and the common ratio ($�$) being 1.05.

You can find the future amount ($�$) using the explicit formula for a geometric sequence:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

In this case, you want to find ${�}_{10}$ (the amount after 10 years):

${�}_{10}=1000\cdot (1.05{)}^{10}$

Calculate this to find the future amount:

${�}_{10}\approx 1628.89$

So, after 10 years, you will have approximately $1,628.89 in the savings account.

Example 2: Bacterial Growth

Suppose a culture of bacteria doubles in size every 4 hours. You start with 10 bacteria. You want to find out how many bacteria will be present after 24 hours.

This problem can be modeled as a geometric sequence where the initial number of bacteria (${�}_1$) is 10, and the common ratio ($�$) is 2 (since they double every 4 hours).

To find the number of bacteria after 24 hours (${�}_6$ since 6 sets of 4 hours make 24 hours), you can use the formula for the nth term of a geometric sequence:

${�}_{�}={�}_1\cdot {�}^{(�-1)}$

Substitute the values:

${�}_6=10\cdot 2^{(6-1)}$

Simplify:

${�}_6=10\cdot 2^5$

${�}_6=10\cdot 32$

${�}_6=320$

So, after 24 hours, you will have 320 bacteria in the culture.

These examples illustrate how geometric sequences can be used to solve application problems involving exponential growth or decay. It's essential to identify the parameters of the geometric sequence and use the appropriate formulas to find the solutions.

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. 1. −5,−11,−24,−47,−97,... 2. 4,5.2,4.4,5.4,5.8,... 3. −1,12,−14,13,−126,... 4. 6,8,11,12,20,... 5. 0.8,4,20,102,520,... For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. 6. a1=8,r=0.3 7. a1=5,r=12

Let's determine whether each sequence is geometric and find the common ratio if it is.

Sequence 1: -5, -11, -24, -47, -97, ...

To check if this is a geometric sequence, we need to see if there is a common ratio between the terms. Let's calculate the ratios between consecutive terms:

• Common Ratio between -11 and -5: $-11\mathrm{/}-5=11\mathrm{/}5$
• Common Ratio between -24 and -11: $-24\mathrm{/}-11=24\mathrm{/}11$
• Common Ratio between -47 and -24: $-47\mathrm{/}-24=47\mathrm{/}24$
• Common Ratio between -97 and -47: $-97\mathrm{/}-47=97\mathrm{/}47$

As you can see, the common ratios are not the same for all pairs of consecutive terms. Therefore, this sequence is not geometric.

Sequence 2: 4, 5.2, 4.4, 5.4, 5.8, ...

Let's calculate the ratios between consecutive terms:

• Common Ratio between 5.2 and 4: $5.2\mathrm{/}4=1.3$
• Common Ratio between 4.4 and 5.2: $4.4\mathrm{/}5.2=0.846153846$
• Common Ratio between 5.4 and 4.4: $5.4\mathrm{/}4.4=1.227272727$
• Common Ratio between 5.8 and 5.4: $5.8\mathrm{/}5.4=1.074074074$

The common ratios are not the same for all pairs of consecutive terms. Therefore, this sequence is not geometric.

Sequence 3: -1, 12, -14, 13, -126, ...

Let's calculate the ratios between consecutive terms:

• Common Ratio between 12 and -1: $12\mathrm{/}-1=-12$
• Common Ratio between -14 and 12: $-14\mathrm{/}12=-1.166666667$
• Common Ratio between 13 and -14: $13\mathrm{/}-14=-0.928571429$
• Common Ratio between -126 and 13: $-126\mathrm{/}13=-9.692307692$

The common ratios are not the same for all pairs of consecutive terms. Therefore, this sequence is not geometric.

Sequence 4: 6, 8, 11, 12, 20, ...

Let's calculate the ratios between consecutive terms:

• Common Ratio between 8 and 6: $8\mathrm{/}6=1.333333333$
• Common Ratio between 11 and 8: $11\mathrm{/}8=1.375$
• Common Ratio between 12 and 11: $12\mathrm{/}11=1.090909091$
• Common Ratio between 20 and 12: $20\mathrm{/}12=1.666666667$

The common ratios are not the same for all pairs of consecutive terms. Therefore, this sequence is not geometric.

Sequence 5: 0.8, 4, 20, 102, 520, ...

Let's calculate the ratios between consecutive terms:

• Common Ratio between 4 and 0.8: $4\mathrm{/}0.8=5$
• Common Ratio between 20 and 4: $20\mathrm{/}4=5$
• Common Ratio between 102 and 20: $102\mathrm{/}20=5.1$
• Common Ratio between 520 and 102: $520\mathrm{/}102=5.098039216$

The common ratios are approximately the same for all pairs of consecutive terms, specifically close to 5. Therefore, this sequence is geometric, and the common ratio is approximately 5.

Now, let's write the first five terms of the geometric sequences given the first term and common ratio:

Sequence 6: ${�}_1=8$, $�=0.3$

• First Term: ${�}_1=8$
• Second Term: ${�}_2={�}_1\cdot �=8\cdot 0.3=2.4$
• Third Term: ${�}_3={�}_2\cdot �=2.4\cdot 0.3=0.72$
• Fourth Term: ${�}_4={�}_3\cdot �=0.72\cdot 0.3=0.216$
• Fifth Term: ${�}_5={�}_4\cdot �=0.216\cdot 0.3=0.0648$

The first five terms of this geometric sequence are 8, 2.4, 0.72, 0.216, and 0.0648.

Sequence 7: ${�}_1=5$, $�=12$

• First Term: ${�}_1=5$
• Second Term: (a_2 = a_1 \cdot r = 5 \cdot 12 = 60
• Third Term: (a_3 = a_2 \cdot r = 60 \cdot 12 = 720
• Fourth Term: (a_4 = a_3 \cdot r = 720 \cdot 12 = 8640
• Fifth Term: (a_5 = a_4 \cdot r = 8640 \cdot 12 = 103680

The first five terms of this geometric sequence are 5, 60, 720, 8640, and 103680.