Write An Equation For The Nth Term Of The Geometric Sequence: $0.6, -3, 15, -75, \ldots$(a) $a_n = \square$

Introduction to 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. In this article, we will focus on writing an equation for the nth term of a given geometric sequence.

Understanding the Given Sequence

The given sequence is: $0.6, -3, 15, -75, \ldots$ To find the common ratio, we can divide each term by the previous term.

• $\frac{-3}{0.6} = -5$
• $\frac{15}{-3} = -5$
• $\frac{-75}{15} = -5$

As we can see, the common ratio is $-5$. This means that each term in the sequence is obtained by multiplying the previous term by $-5$.

Writing an Equation for the nth Term

Now that we have the common ratio, we can write an equation for the nth term of the sequence. The general formula for the nth term of a geometric sequence is:

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

where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $r$ is the common ratio
• $n$ is the term number

In this case, the first term $a_1$ is $0.6$ and the common ratio $r$ is $-5$. Plugging these values into the formula, we get:

$a_n = 0.6 \cdot (-5)^{(n-1)}$

Simplifying the Equation

We can simplify the equation by expanding the exponent:

$a_n = 0.6 \cdot (-5)^{(n-1)}$ $a_n = 0.6 \cdot (-5)^n \cdot (-5)^{-1}$ $a_n = 0.6 \cdot (-5)^n \cdot \frac{1}{-5}$ $a_n = 0.6 \cdot (-5)^n \cdot \frac{-1}{5}$ $a_n = \frac{-0.6 \cdot (-5)^n}{5}$

Final Equation

The final equation for the nth term of the geometric sequence is:

$a_n = \frac{-0.6 \cdot (-5)^n}{5}$

Example Use Case

Let's say we want to find the 5th term of the sequence. We can plug $n = 5$ into the equation:

$a_5 = \frac{-0.6 \cdot (-5)^5}{5}$ $a_5 = \frac{-0.6 \cdot (-3125)}{5}$ $a_5 = \frac{1875}{5}$ $a_5 = 375$

Therefore, the 5th term of the sequence is $375$.

Conclusion

In this article, we learned how to write an equation for the nth term of a geometric sequence. We used the given sequence $0.6, -3, 15, -75, \ldots$ to find the common ratio and then used the general formula for the nth term to write an equation. We simplified the equation and provided an example use case to demonstrate how to find a specific term in the sequence.

Introduction

In our previous article, we discussed how to write an equation for the nth term of a geometric sequence. 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 common ratio of a geometric sequence?

A: To find the common ratio, you can divide each term by the previous term. If the result is the same for each pair of consecutive terms, then that result is the common ratio.

Q: What is the general formula for the nth term of a geometric sequence?

A: The general formula for the nth term of a geometric sequence is:

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

where:

• $a_n$ is the nth term
• $a_1$ is the first term
• $r$ is the common ratio
• $n$ is the term number

Q: How do I write an equation for the nth term of a geometric sequence?

A: To write an equation for the nth term of a geometric sequence, you need to know the first term and the common ratio. You can then plug these values into the general formula for the nth term.

Q: Can I have a negative common ratio?

A: Yes, you can have a negative common ratio. In fact, the common ratio can be any non-zero number, positive or negative.

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

A: The sum of a geometric sequence can be found using the formula:

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

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 have a geometric sequence with a common ratio of 1?

A: Yes, you can have a geometric sequence with a common ratio of 1. In this case, the sequence is an arithmetic sequence, not a geometric sequence.

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 check if the ratio of consecutive terms is constant. If the ratio is constant, then the sequence is geometric. If the ratio is not constant, then the sequence is arithmetic.

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

A: No, you cannot have a geometric sequence with a common ratio of 0. The common ratio must be a non-zero number.

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

A: To find the nth term of a geometric sequence if you know the sum of the first n terms, you can use the formula:

$a_n = \frac{S_n \cdot r}{1 - r}$

where:

• $a_n$ is the nth term
• $S_n$ is the sum of the first n terms
• $r$ is the common ratio

Conclusion

In this article, we answered some frequently asked questions about geometric sequences. We covered topics such as finding the common ratio, writing an equation for the nth term, and determining if a sequence is geometric or arithmetic. We also provided formulas for finding the sum and nth term of a geometric sequence.