Find The Sixth Term Of The Geometric Sequence, Given The First Term And Common Ratio.$a_1 = 5 \text{ And } R = \frac{3}{2}$

Feb 28, 2025 by ADMIN 124 views

**Finding the Sixth Term of a Geometric Sequence**

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Introduction

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this article, we will discuss how to find the sixth term of a geometric sequence given the first term and the common ratio.

Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. The formula for the nth term of a geometric sequence is given by:

a_n = a_1 \cdot r^{(n-1)}

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Given Information

In this problem, we are given the first term $a_1 = 5$ and the common ratio $r = \frac{3}{2}$ . We need to find the sixth term of the geometric sequence.

Finding the Sixth Term

To find the sixth term, we can use the formula for the nth term of a geometric sequence:

a_n = a_1 \cdot r^{(n-1)}

Substituting the given values, we get:

a_6 = 5 \cdot \left(\frac{3}{2}\right)^{(6-1)}

a_6 = 5 \cdot \left(\frac{3}{2}\right)^5

Calculating the Sixth Term

To calculate the sixth term, we need to evaluate the expression $\left(\frac{3}{2}\right)^5$ . We can do this by multiplying the numerator and denominator by the same number:

\left(\frac{3}{2}\right)^5 = \frac{3^5}{2^5}

\left(\frac{3}{2}\right)^5 = \frac{243}{32}

Finding the Sixth Term

Now that we have evaluated the expression, we can substitute it back into the formula for the sixth term:

a_6 = 5 \cdot \frac{243}{32}

a_6 = \frac{1215}{32}

Conclusion

In this article, we discussed how to find the sixth term of a geometric sequence given the first term and the common ratio. We used the formula for the nth term of a geometric sequence and evaluated the expression to find the sixth term. The final answer is $\boxed{\frac{1215}{32}}$ .

Example Problems

Here are a few example problems to help you practice finding the nth term of a geometric sequence:

Find the fifth term of a geometric sequence with first term $a_1 = 2$ and common ratio $r = \frac{4}{3}$ .
Find the eighth term of a geometric sequence with first term $a_1 = 3$ and common ratio $r = \frac{2}{3}$ .
Find the tenth term of a geometric sequence with first term $a_1 = 4$ and common ratio $r = \frac{3}{4}$ .

Tips and Tricks

Here are a few tips and tricks to help you find the nth term of a geometric sequence:

Make sure to use the correct formula for the nth term of a geometric sequence.
Evaluate the expression carefully to avoid errors.
Use a calculator to check your answer if necessary.

Real-World Applications

Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate interest rates and investment returns.
Music: Geometric sequences are used to calculate the frequency of musical notes.
Science: Geometric sequences are used to calculate the growth of populations and the spread of diseases.

Conclusion

In conclusion, finding the nth term of a geometric sequence is a simple process that involves using the formula for the nth term and evaluating the expression. With practice and patience, you can become proficient in finding the nth term of a geometric sequence and apply it to real-world problems.

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Introduction

In our previous article, we discussed how to find the sixth term of a geometric sequence given the first term and the common ratio. In this article, we will answer some frequently asked questions about geometric sequences.

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do I find the nth term of a geometric sequence?

A: To find the nth term of a geometric sequence, you can use the formula:

a_n = a_1 \cdot r^{(n-1)}

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What is the common ratio?

A: The common ratio is a fixed, non-zero number that is used to find each term of a geometric sequence. It is the ratio of any term to its previous term.

Q: How do I find the common ratio?

A: To find the common ratio, you can divide any term by its previous term. For example, if the first term is 5 and the second term is 15, the common ratio is 15/5 = 3.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is:

S_n = \frac{a_1(1-r^n)}{1-r}

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I find the sum of a geometric sequence?

A: To find the sum of a geometric sequence, you can use the formula:

S_n = \frac{a_1(1-r^n)}{1-r}

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the formula for the nth partial sum of a geometric sequence?

A: The formula for the nth partial sum of a geometric sequence is:

S_n = \frac{a_1(1-r^n)}{1-r}

where $S_n$ is the nth partial sum, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I find the nth partial sum of a geometric sequence?

A: To find the nth partial sum of a geometric sequence, you can use the formula:

S_n = \frac{a_1(1-r^n)}{1-r}

where $S_n$ is the nth partial sum, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the formula for the product of a geometric sequence?

A: The formula for the product of a geometric sequence is:

P_n = a_1 \cdot r^{(n-1)}

where $P_n$ is the product of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I find the product of a geometric sequence?

A: To find the product of a geometric sequence, you can use the formula:

P_n = a_1 \cdot r^{(n-1)}

where $P_n$ is the product of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Conclusion

In conclusion, geometric sequences are a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. We have answered some frequently asked questions about geometric sequences, including how to find the nth term, the common ratio, the sum, the nth partial sum, and the product of a geometric sequence.

Example Problems

Here are a few example problems to help you practice finding the nth term, the common ratio, the sum, the nth partial sum, and the product of a geometric sequence:

Find the fifth term of a geometric sequence with first term $a_1 = 2$ and common ratio $r = \frac{4}{3}$ .
Find the eighth term of a geometric sequence with first term $a_1 = 3$ and common ratio $r = \frac{2}{3}$ .
Find the tenth term of a geometric sequence with first term $a_1 = 4$ and common ratio $r = \frac{3}{4}$ .

Tips and Tricks

Here are a few tips and tricks to help you find the nth term, the common ratio, the sum, the nth partial sum, and the product of a geometric sequence:

Make sure to use the correct formula for the nth term, the common ratio, the sum, the nth partial sum, and the product of a geometric sequence.
Evaluate the expression carefully to avoid errors.
Use a calculator to check your answer if necessary.

Real-World Applications

Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate interest rates and investment returns.
Music: Geometric sequences are used to calculate the frequency of musical notes.
Science: Geometric sequences are used to calculate the growth of populations and the spread of diseases.