Given The Explicit Formula For A Geometric Sequence, Find The 8th Term.$ A_n = 3^{n-1} }$Options A. { A_8 = 2187 $ $B. { A_8 = 2000 $}$C. { A_8 = 3 $}$D. { A_8 = 1 $}$

Mar 12, 2025 by ADMIN 169 views

Finding the 8th Term of a Geometric Sequence

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, Find the 8th Term

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 the sequence. To do this, we need to substitute ${ n = 8 }$ into the formula and simplify.

Substituting n = 8 into the Formula

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

${ a_8 = 3^7 }$

Evaluating the Expression

To evaluate the expression ${ 3^7 }$ , we need to multiply 3 by itself 7 times:

${ 3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 }$

${ 3^7 = 2187 }$

Conclusion

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

Answer

The correct answer is:

${ a_8 = 2187 }$

Explanation

The other options are incorrect because they do not match the result of substituting ${ n = 8 }$ into the formula and simplifying.

Option B is incorrect because ${ 3^7 }$ is not equal to 2000.
Option C is incorrect because ${ 3^7 }$ is not equal to 3.
Option D is incorrect because ${ 3^7 }$ is not equal to 1.

Example Use Case

This problem can be used to demonstrate the concept of geometric sequences and how to find the nth term using the explicit formula. It can also be used to practice evaluating expressions with exponents.

Tips and Tricks

When substituting values into the formula, make sure to follow the order of operations (PEMDAS).
When evaluating expressions with exponents, make sure to multiply the base by itself the correct number of times.
When simplifying expressions, make sure to combine like terms and eliminate any unnecessary parentheses.
Geometric Sequences Q&A

Understanding Geometric Sequences

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

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&A

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is:

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

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

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

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 value of n into the formula and simplify.

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

A: A geometric sequence is a type of sequence where each term is found by multiplying the previous term by a fixed number, while an arithmetic sequence is a type of sequence where each term is found by adding a fixed number to the previous term.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0, because this would result in a sequence where each term is 0.

Q: Can a geometric sequence have a common ratio of 1?

A: Yes, a geometric sequence can have a common ratio of 1, but this would result in a sequence where each term is the same as the previous term.

Q: How do I determine if a sequence is geometric or arithmetic?

A: To determine if a sequence is geometric or arithmetic, you need to look at the pattern of the sequence. If the sequence is formed by multiplying each term by a fixed number, it is a geometric sequence. If the sequence is formed by adding a fixed number to each term, it is an arithmetic sequence.

Q: Can a geometric sequence have a negative common ratio?

A: Yes, a geometric sequence can have a negative common ratio. This would result in a sequence where each term is the negative of the previous term.

Q: Can a geometric sequence have a common ratio that is a fraction?

A: Yes, a geometric sequence can have a common ratio that is a fraction. This would result in a sequence where each term is the fraction multiplied by the previous term.

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 for the sum of a geometric series:

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

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 a geometric sequence have an infinite number of terms?

A: Yes, a geometric sequence can have an infinite number of terms. This would result in a sequence that goes on forever.

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

A: To find the limit of a geometric sequence, you need to use the formula for the limit of a geometric sequence:

${ \lim_{n\to\infty} a_n = \frac{a_1}{1-r} }$

where:

${ a_n }$ is the nth term
${ a_1 }$ is the first term
${ r }$ is the common ratio

Conclusion

Geometric sequences are an important concept in mathematics, and understanding how to work with them is crucial for solving problems in a variety of fields. By following the formula and using the correct techniques, you can find the nth term, sum, and limit of a geometric sequence.