Given the Explicit Formula

Oct 13, 2025 by windailys.com 44 views

Understanding Geometric Sequences

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. The explicit formula for a geometric sequence is given by:

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

where:

${ a_n }$ is the nth term of the sequence
${ a_1 }$ is the first term of the sequence
${ r }$ is the common ratio
${ n }$ is the term number

Given the Explicit Formula

In this problem, we are given the explicit formula for a geometric sequence:

${ a_n = 3^{n-1} }$

We are asked to find the 8th term of this sequence.

Finding the 8th Term

To find the 8th term, we need to substitute ${ n = 8 }$ into the explicit formula:

${ a_8 = 3^{8-1} }$

${ a_8 = 3^7 }$

${ a_8 = 2187 }$

Conclusion

Therefore, the 8th term of the geometric sequence is ${ a_8 = 2187 }$ .

Why is this the Correct Answer?

This is the correct answer because we have correctly substituted ${ n = 8 }$ into the explicit formula and evaluated the expression to get the 8th term.

Common Mistakes to Avoid

One common mistake to avoid is to substitute ${ n = 8 }$ into the formula as ${ a_8 = 3^{8} }$ instead of ${ a_8 = 3^{8-1} }$ . This would result in an incorrect answer.

Real-World Applications

Geometric sequences have many real-world applications, such as:

Finance: Geometric sequences can be used to model the growth of investments over time.
Biology: Geometric sequences can be used to model the growth of populations over time.
Computer Science: Geometric sequences can be used to model the growth of data in computer systems.

Example Problems

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

Find the 5th term of the geometric sequence: ${ a_n = 2^{n-1} }$
Find the 10th term of the geometric sequence: ${ a_n = 5^{n-1} }$

Solutions

${ a_5 = 2^{5-1} = 2^4 = 16 }$
${ a_{10} = 5^{10-1} = 5^9 = 1953125 }$

Conclusion

In conclusion, finding the nth term of a geometric sequence is a straightforward process that involves substituting the term number into the explicit formula and evaluating the expression. With practice, you can become proficient in finding the nth term of a geometric sequence and apply it to real-world problems.

References

[1] "Geometric Sequences" by Math Open Reference
[2] "Geometric Sequences" by Khan Academy

Additional Resources

[1] "Geometric Sequences" by Wolfram MathWorld
[2] "Geometric Sequences" by MIT OpenCourseWare

Final Thoughts

Frequently Asked Questions

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: What is the explicit formula for a geometric sequence?

A: The explicit formula for a geometric sequence is given by:

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

where:

${ a_n }$ is the nth term of the sequence
${ a_1 }$ is the first term of the sequence
${ r }$ is the common ratio
${ n }$ is the term number

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

A: To find the nth term of a geometric sequence, you need to substitute the term number into the explicit formula and evaluate the expression.

Q: What is the common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is the fixed, non-zero number that is multiplied by each term to get the next term.

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

A: To find the common ratio of a geometric sequence, you need to divide any term by the previous term.

Q: What is the difference between an arithmetic sequence and a geometric sequence?

A: An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed number to the previous term. 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.

Q: When would I use a geometric sequence in real life?

A: You would use a geometric sequence in real life when modeling the growth of investments, populations, or data over time.

Q: Can I use a geometric sequence to model a situation where the growth rate is not constant?

A: No, a geometric sequence is only used to model situations where the growth rate is constant.

Q: How do I determine if a sequence is geometric or not?

A: To determine if a sequence is geometric or not, you need to check if the ratio of each term to the previous term is constant.

Q: What are some common mistakes to avoid when working with geometric sequences?

A: Some common mistakes to avoid when working with geometric sequences include:

Substituting the wrong term number into the explicit formula
Not evaluating the expression correctly
Not checking if the sequence is geometric or not

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

A: To find the sum of a geometric sequence, you need to 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
${ n }$ is the number of terms

Q: Can I use a geometric sequence to model a situation where the growth rate is negative?

A: Yes, you can use a geometric sequence to model a situation where the growth rate is negative. In this case, the common ratio would be a negative number.

Conclusion

In conclusion, geometric sequences are an important concept in mathematics that has many real-world applications. By understanding the explicit formula, common ratio, and how to find the nth term, you can use geometric sequences to model and solve problems in finance, biology, computer science, and other fields.

References

[1] "Geometric Sequences" by Math Open Reference
[2] "Geometric Sequences" by Khan Academy

Additional Resources

[1] "Geometric Sequences" by Wolfram MathWorld
[2] "Geometric Sequences" by MIT OpenCourseWare

Final Thoughts

Geometric sequences are a powerful tool for modeling and solving problems in many fields. By understanding the concepts and formulas, you can use geometric sequences to make predictions, analyze data, and make informed decisions.