Write An Explicit Formula For { A_n $}$, The { N $} T H T E R M O F T H E S E Q U E N C E 19 , 13 , 7 , A N S W E R A T T E M P T 1 O U T O F 2 : Th Term Of The Sequence 19, 13, 7, Answer Attempt 1 Out Of 2: T H T Er M O F T H Ese Q U E N Ce 19 , 13 , 7 , A N S W Er A Tt E M Pt 1 O U T O F 2 : {$ A_n = $}$ [Enter Your Formula Here]Submit Answer

Introduction

In this article, we will explore the concept of an explicit formula for a geometric sequence. 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 general form of a geometric sequence is given by:

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

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

Understanding the Given Sequence

The given sequence is 19, 13, 7, ... . To find the explicit formula for this sequence, we need to identify the common ratio. We can do this by dividing each term by the previous term.

$$\frac{13}{19} = \frac{7}{13}$$

Simplifying the above equation, we get:

$$\frac{13}{19} = \frac{7}{13}$$

$$13^2 = 19 \cdot 7$$

$$169 = 133$$

This equation is not true, which means that the given sequence is not a geometric sequence. However, we can try to find a different type of sequence that matches the given sequence.

Finding the Explicit Formula

Let's assume that the given sequence is an arithmetic sequence. 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 to the previous term. The general form of an arithmetic sequence is given by:

$$a_n = a_1 + (n-1)d$$

where $a_n$ is the $n$th term of the sequence, $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

We can use the given sequence to find the common difference. We can do this by subtracting each term from the previous term.

$$13 - 19 = -6$$

$$7 - 13 = -6$$

Since the common difference is the same, we can conclude that the given sequence is an arithmetic sequence.

Calculating the Common Difference

Now that we have identified the sequence as an arithmetic sequence, we can calculate the common difference. We can do this by subtracting each term from the previous term.

$$d = 13 - 19 = -6$$

$$d = 7 - 13 = -6$$

Since the common difference is the same, we can conclude that the common difference is indeed -6.

Finding the Explicit Formula

Now that we have identified the sequence as an arithmetic sequence and calculated the common difference, we can find the explicit formula for the sequence. We can do this by substituting the values of $a_1$ and $d$ into the general form of an arithmetic sequence.

$$a_n = a_1 + (n-1)d$$

$$a_n = 19 + (n-1)(-6)$$

Simplifying the above equation, we get:

$$a_n = 19 - 6n + 6$$

$$a_n = 25 - 6n$$

Conclusion

In this article, we have explored the concept of an explicit formula for a geometric sequence. We have also identified the given sequence as an arithmetic sequence and calculated the common difference. Finally, we have found the explicit formula for the sequence using the general form of an arithmetic sequence.

Explicit Formula for the Sequence

The explicit formula for the sequence is given by:

$$a_n = 25 - 6n$$

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

Example Use Case

Suppose we want to find the 5th term of the sequence. We can use the explicit formula to do this.

$$a_5 = 25 - 6(5)$$

$$a_5 = 25 - 30$$

$$a_5 = -5$$

Therefore, the 5th term of the sequence is -5.

Conclusion

In this article, we have explored the concept of an explicit formula for a geometric sequence. We have also identified the given sequence as an arithmetic sequence and calculated the common difference. Finally, we have found the explicit formula for the sequence using the general form of an arithmetic sequence. The explicit formula for the sequence is given by:

$$a_n = 25 - 6n$$

$$$$

Introduction

In our previous article, we explored the concept of an explicit formula for a geometric sequence. We also identified the given sequence as an arithmetic sequence and calculated the common difference. Finally, we found the explicit formula for the sequence using the general form of an arithmetic sequence. In this article, we will answer some frequently asked questions about explicit formulas for geometric sequences.

Q: What is an explicit formula for a geometric sequence?

A: An explicit formula for a geometric sequence is a mathematical expression that gives the $n$th term of the sequence for any value of $n$. It is typically written in the form:

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

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

Q: How do I find the explicit formula for a geometric sequence?

A: To find the explicit formula for a geometric sequence, you need to identify the common ratio and the first term. You can do this by dividing each term by the previous term. Once you have identified the common ratio and the first term, you can substitute these values into the general form of a geometric sequence.

Q: What is the difference between an explicit formula and an implicit formula?

A: An explicit formula is a mathematical expression that gives the $n$th term of the sequence for any value of $n$. An implicit formula, on the other hand, is a mathematical expression that gives the $n$th term of the sequence in terms of the previous term. For example, the explicit formula for a geometric sequence is:

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

while the implicit formula is:

$$a_n = a_{n-1} \cdot r$$

Q: Can I use an explicit formula to find the $n$th term of an arithmetic sequence?

A: No, you cannot use an explicit formula to find the $n$th term of an arithmetic sequence. An explicit formula is typically used to find the $n$th term of a geometric sequence, while an arithmetic sequence requires a different type of formula.

Q: How do I use an explicit formula to find the $n$th term of a geometric sequence?

A: To use an explicit formula to find the $n$th term of a geometric sequence, you need to substitute the values of $a_1$ and $r$ into the general form of a geometric sequence. For example, if the first term is 2 and the common ratio is 3, the explicit formula would be:

$$a_n = 2 \cdot 3^{(n-1)}$$

You can then use this formula to find the $n$th term of the sequence for any value of $n$.

Q: What are some common mistakes to avoid when finding an explicit formula for a geometric sequence?

A: Some common mistakes to avoid when finding an explicit formula for a geometric sequence include:

• Not identifying the common ratio correctly
• Not substituting the values of $a_1$ and $r$ into the general form of a geometric sequence
• Not simplifying the formula correctly
• Not using the correct formula for the $n$th term of a geometric sequence

Conclusion

In this article, we have answered some frequently asked questions about explicit formulas for geometric sequences. We have also provided some tips and tricks for finding explicit formulas and avoiding common mistakes. By following these tips and tricks, you can become more confident and proficient in finding explicit formulas for geometric sequences.