Which Formula Can Be Used To Find The $n$th Term Of A Geometric Sequence Where The Fifth Term Is $\frac{1}{16}$ And The Common Ratio Is $\frac{1}{4}$?A. $a_n = 16\left(\frac{1}{4}\right)^{n-1}$B. $a_n =

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 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

We are given that the fifth term of a geometric sequence is $\frac{1}{16}$ and the common ratio is $\frac{1}{4}$. We need to find the formula for the nth term of this sequence.

Finding the First Term

To find the formula for the nth term, we need to find the first term $a_1$. We can use the given information to do this. Since the fifth term is $\frac{1}{16}$, we can write:

$$a_5 = a_1 \cdot r^{5-1} = a_1 \cdot \left(\frac{1}{4}\right)^4 = a_1 \cdot \frac{1}{256}$$

We are also given that the fifth term is $\frac{1}{16}$, so we can set up the equation:

$$a_1 \cdot \frac{1}{256} = \frac{1}{16}$$

Solving for $a_1$, we get:

$$a_1 = \frac{1}{16} \cdot \frac{256}{1} = 16$$

Finding the Formula for the nth Term

Now that we have found the first term $a_1$, we can find the formula for the nth term. We can use the formula:

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

Substituting the values we have found, we get:

$$a_n = 16 \cdot \left(\frac{1}{4}\right)^{n-1}$$

Conclusion

In conclusion, the formula for the nth term of a geometric sequence where the fifth term is $\frac{1}{16}$ and the common ratio is $\frac{1}{4}$ is:

$$a_n = 16 \cdot \left(\frac{1}{4}\right)^{n-1}$$

This formula can be used to find any term of the sequence, given the value of $n$.

Example

Let's use the formula to find the 10th term of the sequence.

$$a_{10} = 16 \cdot \left(\frac{1}{4}\right)^{10-1} = 16 \cdot \left(\frac{1}{4}\right)^9 = 16 \cdot \frac{1}{262144} = \frac{1}{16384}$$

Discussion

The formula for the nth term of a geometric sequence is a powerful tool for finding any term of the sequence, given the value of $n$. In this example, we used the formula to find the 10th term of a sequence where the fifth term is $\frac{1}{16}$ and the common ratio is $\frac{1}{4}$.

Common Ratio

The common ratio is a key component of a geometric sequence. It determines how each term is related to the previous term. In this example, the common ratio is $\frac{1}{4}$, which means that each term is one-fourth of the previous term.

Geometric Sequences in Real Life

Geometric sequences have many real-life applications. They can be used to model population growth, financial investments, and many other phenomena. In this example, we used a geometric sequence to model a situation where the fifth term is $\frac{1}{16}$ and the common ratio is $\frac{1}{4}$.

Conclusion

In conclusion, the formula for the nth term of a geometric sequence is a powerful tool for finding any term of the sequence, given the value of $n$. We used this formula to find the 10th term of a sequence where the fifth term is $\frac{1}{16}$ and the common ratio is $\frac{1}{4}$. Geometric sequences have many real-life applications and are an important concept in mathematics.

Final Answer

The final answer is: $\boxed{16 \cdot \left(\frac{1}{4}\right)^{n-1}}$

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 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.

Q&A

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

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

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

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

A: The common ratio is a fixed, non-zero number that is used to find each term of the sequence. It is denoted by the letter $r$.

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

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

$$a_1 = \frac{a_n}{r^{n-1}}$$

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

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 determine if a sequence is geometric or arithmetic?

A: To determine if a sequence is geometric or arithmetic, you can look at the relationship between the terms. If the terms are related by a fixed, non-zero number (the common ratio), then the sequence is geometric. If the terms are related by a fixed number (the common difference), then the sequence is arithmetic.

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

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

• Modeling population growth
• Financial investments
• Compound interest
• Music and art

Q: How do I use a geometric sequence to model population growth?

A: To use a geometric sequence to model population growth, you can use the formula:

$$P_n = P_1 \cdot r^{n-1}$$

where $P_n$ is the population at time $n$, $P_1$ is the initial population, and $r$ is the growth rate.

Q: How do I use a geometric sequence to model financial investments?

A: To use a geometric sequence to model financial investments, you can use the formula:

$$A_n = A_1 \cdot r^{n-1}$$

where $A_n$ is the amount of money at time $n$, $A_1$ is the initial amount, and $r$ is the interest rate.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling many real-life phenomena. By understanding the formula for the nth term of a geometric sequence, you can use it to model population growth, financial investments, and many other applications.

Final Answer

The final answer is: $\boxed{16 \cdot \left(\frac{1}{4}\right)^{n-1}}$