If $a_1=6$ And $a_n=5 A_{n-1}$, Then Find The Value Of \$a_6$[/tex\].

**Understanding the Recursive Sequence** =====================================

Introduction

In mathematics, a recursive sequence is a sequence of numbers where each term is defined recursively as a function of the preceding term. In this article, we will explore a specific recursive sequence and find the value of its sixth term.

The Recursive Sequence

The given recursive sequence is defined as follows:

$$a_1 = 6$$

$$a_n = 5a_{n-1}$$

where $a_n$ is the nth term of the sequence.

Finding the Value of $a_6$

To find the value of $a_6$, we need to find the values of $a_2$, $a_3$, $a_4$, and $a_5$ first.

Finding $a_2$

Using the recursive formula, we can find the value of $a_2$ as follows:

$$a_2 = 5a_1 = 5(6) = 30$$

Finding $a_3$

Now that we have the value of $a_2$, we can find the value of $a_3$ as follows:

$$a_3 = 5a_2 = 5(30) = 150$$

Finding $a_4$

Using the value of $a_3$, we can find the value of $a_4$ as follows:

$$a_4 = 5a_3 = 5(150) = 750$$

Finding $a_5$

Now that we have the value of $a_4$, we can find the value of $a_5$ as follows:

$$a_5 = 5a_4 = 5(750) = 3750$$

Finding $a_6$

Finally, we can find the value of $a_6$ as follows:

$$a_6 = 5a_5 = 5(3750) = 18750$$

Conclusion

In this article, we explored a recursive sequence and found the value of its sixth term. We used the recursive formula to find the values of $a_2$, $a_3$, $a_4$, and $a_5$ first, and then used these values to find the value of $a_6$.

Q&A

Q: What is a recursive sequence?

A: A recursive sequence is a sequence of numbers where each term is defined recursively as a function of the preceding term.

Q: How do we find the value of a term in a recursive sequence?

A: We use the recursive formula to find the value of each term. The recursive formula is given by $a_n = f(a_{n-1})$, where $f$ is a function that takes the previous term as input and returns the current term.

Q: What is the value of $a_6$ in the given recursive sequence?

A: The value of $a_6$ is 18750.

Q: How do we find the value of $a_6$ in the given recursive sequence?

A: We find the value of $a_6$ by first finding the values of $a_2$, $a_3$, $a_4$, and $a_5$ using the recursive formula, and then using these values to find the value of $a_6$.

Q: What is the recursive formula for the given sequence?

A: The recursive formula for the given sequence is $a_n = 5a_{n-1}$.

Q: What is the initial term of the given sequence?

A: The initial term of the given sequence is $a_1 = 6$.

Q: How many terms do we need to find to find the value of $a_6$?

A: We need to find the values of $a_2$, $a_3$, $a_4$, and $a_5$ to find the value of $a_6$.