Calculating the Eighth Term

Oct 13, 2025 by windailys.com 121 views

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 eighth term of a geometric sequence given the first term and the common ratio.

What is a Geometric Sequence?

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 $a_1 = 2$ and the common ratio $r = -2$ . We need to find the eighth term of the geometric sequence.

Formula for the nth Term

Using the formula for the nth term of a geometric sequence, we can plug in the given values to find the eighth term:

a_8 = a_1 \cdot r^{(8-1)}

a_8 = 2 \cdot (-2)^7

Calculating the Eighth Term

To calculate the eighth term, we need to evaluate the expression $(-2)^7$ . This can be done using the following steps:

Raise -2 to the power of 7:

(-2)^7 = -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2

Simplify the expression:

(-2)^7 = -128

Now, we can plug this value back into the expression for the eighth term:

a_8 = 2 \cdot (-128)

a_8 = -256

Conclusion

In this article, we discussed how to find the eighth 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 plugged in the given values to find the eighth term. The final answer is $a_8 = -256$ .

Example Use Case

This problem can be used as an example in a mathematics class to teach students how to find the nth term of a geometric sequence. It can also be used to demonstrate the importance of following the order of operations when evaluating expressions.

Tips and Variations

To find the nth term of a geometric sequence, you can use the formula $a_n = a_1 \cdot r^{(n-1)}$ .
If the common ratio is positive, the sequence will increase exponentially. If the common ratio is negative, the sequence will decrease exponentially.
You can also use this formula to find the sum of the first n terms of a geometric sequence.

Common Mistakes

Make sure to follow the order of operations when evaluating expressions.
Make sure to plug in the correct values into the formula.
Make sure to simplify the expression before evaluating it.

References

[1] "Geometric Sequences and Series". Math Open Reference.
[2] "Formula for the nth Term of a Geometric Sequence". Math Is Fun.
Geometric Sequence Q&A =========================

Frequently Asked Questions

In this article, we will answer some of the most 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 denoted by the letter $r$ .

Q: How do I find the common ratio?

A: To find the common ratio, you can use the formula $r = \frac{a_n}{a_{n-1}}$ , where $a_n$ is the nth term and $a_{n-1}$ is the (n-1)th term.

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 after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

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

A: To find the sum of the first n terms 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 sum of an infinite geometric sequence?

A: The formula for the sum of an infinite geometric sequence is $S = \frac{a_1}{1-r}$ , where $S$ is the sum, $a_1$ is the first term, and $r$ is the common ratio.

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

A: To determine if a sequence is geometric, you can check if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What are some real-world applications of geometric sequences?

A: Geometric sequences have many real-world applications, including:

Compound interest: Geometric sequences can be used to calculate the future value of an investment.
Population growth: Geometric sequences can be used to model the growth of a population.
Music: Geometric sequences can be used to create musical patterns and rhythms.

Q: How do I use geometric sequences in real-world problems?

A: To use geometric sequences in real-world problems, you can use the formulas and concepts discussed in this article to model and solve problems.

Conclusion

In this article, we have answered some of the most frequently asked questions about geometric sequences. We hope that this article has been helpful in understanding the concepts and formulas of geometric sequences.

Example Use Case

This article can be used as a reference for students and professionals who need to understand geometric sequences and their applications.

Tips and Variations

To find the nth term of a geometric sequence, you can use the formula $a_n = a_1 \cdot r^{(n-1)}$ .
To find the sum of the first n terms of a geometric sequence, you can use the formula $S_n = \frac{a_1(1-r^n)}{1-r}$ .
To find the sum of an infinite geometric sequence, you can use the formula $S = \frac{a_1}{1-r}$ .

Common Mistakes

Make sure to follow the order of operations when evaluating expressions.
Make sure to plug in the correct values into the formulas.
Make sure to simplify the expressions before evaluating them.

References

[1] "Geometric Sequences and Series". Math Open Reference.
[2] "Formula for the nth Term of a Geometric Sequence". Math Is Fun.