Several Terms Of A Sequence { \left{a_n\right}_ N=1}^{\infty}$}$ Are Given Below { \left{1, \frac{1 {6}, \frac{1}{36}, \frac{1}{216}, \frac{1}{1,296}, \ldots\right}$}$ A. Find The Next Two Terms Of The Sequence. [$a_6 =

Introduction

In mathematics, a sequence is a list of numbers in a specific order. Sequences can be defined by a formula, and each term in the sequence is obtained by applying the formula to the previous term. In this article, we will explore a given sequence and find the next two terms.

The Given Sequence

The given sequence is:

$$\left\{1, \frac{1}{6}, \frac{1}{36}, \frac{1}{216}, \frac{1}{1296}, \ldots\right\}$$

This sequence appears to be a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant.

Understanding the Sequence

To understand the sequence, let's examine the first few terms:

• $a_1 = 1$
• $a_2 = \frac{1}{6} = a_1 \times \frac{1}{6}$
• $a_3 = \frac{1}{36} = a_2 \times \frac{1}{6}$
• $a_4 = \frac{1}{216} = a_3 \times \frac{1}{6}$
• $a_5 = \frac{1}{1296} = a_4 \times \frac{1}{6}$

We can see that each term is obtained by multiplying the previous term by $\frac{1}{6}$.

Finding the Next Two Terms

To find the next two terms, we can continue the pattern:

• $a_6 = a_5 \times \frac{1}{6} = \frac{1}{1296} \times \frac{1}{6} = \frac{1}{7776}$
• $a_7 = a_6 \times \frac{1}{6} = \frac{1}{7776} \times \frac{1}{6} = \frac{1}{46656}$

Therefore, the next two terms of the sequence are:

$$a_6 = \frac{1}{7776}$$

$$a_7 = \frac{1}{46656}$$

Conclusion

In this article, we explored a given sequence and found the next two terms. We used the pattern of the sequence to determine the next two terms, which are $\frac{1}{7776}$ and $\frac{1}{46656}$. This demonstrates the importance of understanding the pattern of a sequence in order to find its next terms.

Understanding Geometric Sequences

Geometric sequences are a type of sequence where each term is obtained by multiplying the previous term by a fixed constant. The formula for a geometric sequence is:

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

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

In the given sequence, the common ratio is $\frac{1}{6}$. Therefore, the formula for the sequence is:

$$a_n = 1 \times \left(\frac{1}{6}\right)^{n-1}$$

This formula can be used to find any term in the sequence.

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 algorithms over time.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling the growth of quantities over time. By understanding the pattern of a geometric sequence, we can find its next terms and use it to model real-world phenomena.

References

• Khan Academy: Geometric Sequences
• Math Is Fun: Geometric Sequences
• Wikipedia: Geometric Sequence
  Q&A: Geometric Sequences ==========================

Introduction

In our previous article, we explored a given sequence and found the next two terms. We also discussed the concept of geometric sequences and their real-world applications. 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 is obtained by multiplying the previous term by a fixed constant. The formula for a geometric sequence is:

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

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 in a geometric sequence?

A: The common ratio in a geometric sequence is the fixed constant that is multiplied by each term to obtain the next term. In the given sequence, the common ratio is $\frac{1}{6}$.

Q: How do I find the next term in a geometric sequence?

A: To find the next term in a geometric sequence, you can use the formula:

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

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 formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is:

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

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 nth term of a geometric sequence?

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

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

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: How do I determine if a sequence is geometric?

A: To determine if a sequence is geometric, you can look for a pattern where each term is obtained by multiplying the previous term by a fixed constant. You can also use the formula:

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

to see if it fits the sequence.

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

A: 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 algorithms over time.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling the growth of quantities over time. By understanding the pattern of a geometric sequence, we can find its next terms and use it to model real-world phenomena.

References

• Khan Academy: Geometric Sequences
• Math Is Fun: Geometric Sequences
• Wikipedia: Geometric Sequence