The First Three Terms Of A Sequence Are Given. Write Your Answer As A Decimal Or Whole Number. Round To The Nearest Thousandth If Necessary.Sequence: 1 , 5 2 , 25 4 , … 1, \frac{5}{2}, \frac{25}{4}, \ldots 1 , 2 5 ​ , 4 25 ​ , … Find The 6th Term. □ \square □

Understanding the Sequence

The given sequence is $1, \frac{5}{2}, \frac{25}{4}, \ldots$. To find the 6th term, we need to identify the pattern or rule that governs the sequence. The first three terms are provided, and we can use them to determine the common ratio or the formula that generates each term.

Identifying the Pattern

Upon examining the sequence, we notice that each term is obtained by multiplying the previous term by a certain value. Let's calculate the ratio between consecutive terms:

• $\frac{\frac{5}{2}}{1} = \frac{5}{2}$
• $\frac{\frac{25}{4}}{\frac{5}{2}} = \frac{5}{2}$

The common ratio between consecutive terms is $\frac{5}{2}$. This indicates that the sequence is a geometric progression (GP) with a common ratio of $\frac{5}{2}$.

Formula for the nth Term

For a geometric progression, the nth term can be found using 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.

In this case, $a_1 = 1$, $r = \frac{5}{2}$, and we want to find the 6th term, so $n = 6$. Plugging these values into the formula, we get:

$a_6 = 1 \cdot \left(\frac{5}{2}\right)^{(6-1)}$ $a_6 = 1 \cdot \left(\frac{5}{2}\right)^5$ $a_6 = \frac{5^5}{2^5}$

Calculating the 6th Term

Now, let's calculate the value of the 6th term:

$a_6 = \frac{5^5}{2^5}$ $a_6 = \frac{3125}{32}$ $a_6 = 97.65625$

Rounding to the nearest thousandth, the 6th term of the sequence is approximately $\boxed{97.656}$.

Conclusion

In this problem, we were given the first three terms of a sequence and asked to find the 6th term. By identifying the common ratio and using the formula for the nth term of a geometric progression, we were able to calculate the value of the 6th term. The result is a decimal value, rounded to the nearest thousandth.

Key Takeaways

• The sequence is a geometric progression with a common ratio of $\frac{5}{2}$.
• The formula for the nth term of a geometric progression is $a_n = a_1 \cdot r^{(n-1)}$.
• The 6th term of the sequence is approximately $\boxed{97.656}$.

Further Exploration

This problem can be extended by exploring other properties of geometric progressions, such as the sum of the first n terms or the relationship between the terms and the common ratio. Additionally, you can try finding the 10th term or the 15th term of the sequence using the same formula.

Understanding the Sequence

The given sequence is $1, \frac{5}{2}, \frac{25}{4}, \ldots$. To find the 6th term, we need to identify the pattern or rule that governs the sequence. The first three terms are provided, and we can use them to determine the common ratio or the formula that generates each term.

Q: What is the common ratio of the sequence?

A: The common ratio between consecutive terms is $\frac{5}{2}$. This indicates that the sequence is a geometric progression (GP) with a common ratio of $\frac{5}{2}$.

Q: How do I find the nth term of a geometric progression?

A: For a geometric progression, the nth term can be found using 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 formula for the nth term of a geometric progression?

A: The formula for the nth term of a geometric progression is $a_n = a_1 \cdot r^{(n-1)}$. In this case, $a_1 = 1$, $r = \frac{5}{2}$, and we want to find the 6th term, so $n = 6$.

Q: How do I calculate the 6th term of the sequence?

A: To calculate the 6th term, we plug the values into the formula:

$a_6 = 1 \cdot \left(\frac{5}{2}\right)^{(6-1)}$ $a_6 = 1 \cdot \left(\frac{5}{2}\right)^5$ $a_6 = \frac{5^5}{2^5}$

Q: What is the value of the 6th term of the sequence?

A: The value of the 6th term of the sequence is approximately $\boxed{97.656}$.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

• The sequence is a geometric progression with a common ratio of $\frac{5}{2}$.
• The formula for the nth term of a geometric progression is $a_n = a_1 \cdot r^{(n-1)}$.
• The 6th term of the sequence is approximately $\boxed{97.656}$.

Q: How can I further explore this problem?

A: You can further explore this problem by:

• Exploring other properties of geometric progressions, such as the sum of the first n terms or the relationship between the terms and the common ratio.
• Trying to find the 10th term or the 15th term of the sequence using the same formula.

Conclusion

In this Q&A article, we have explored the sequence $1, \frac{5}{2}, \frac{25}{4}, \ldots$ and found the 6th term using the formula for the nth term of a geometric progression. We have also answered some common questions related to this problem and provided some key takeaways and suggestions for further exploration.