Write An Explicit Formula For $a_n$, The $n^{\text{th}}$ Term Of The Sequence \$19, 13, 7, \ldots$[/tex\].$a_n =$ $\square$

Feb 27, 2025 by ADMIN 129 views

**Explicit Formula for the nth Term of a Geometric Sequence**

Introduction

In mathematics, a sequence is a list of numbers in a specific order. A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed constant, known as the common ratio. In this article, we will derive an explicit formula for the nth term of a geometric sequence, given by the sequence 19, 13, 7, ...

Understanding the Sequence

The given sequence is 19, 13, 7, ... . To find the explicit formula for the nth term, we need to identify the common ratio between consecutive terms. The common ratio is calculated by dividing each term by its previous term.

To find the common ratio between the first and second term, we divide 13 by 19, which gives us a common ratio of 13/19.
To find the common ratio between the second and third term, we divide 7 by 13, which gives us a common ratio of 7/13.

Since the common ratio is not the same in both cases, we need to re-examine the sequence. However, if we look closely, we can see that each term is obtained by subtracting 6 from the previous term.

19 - 6 = 13
13 - 6 = 7
7 - 6 = -1 (this is the next term in the sequence)

So, the common ratio is actually -6/1 = -6.

Deriving the Explicit Formula

Now that we have identified the common ratio, we can derive the explicit formula for the nth term of the sequence. 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.

In this case, the first term $a_1$ is 19, and the common ratio $r$ is -6. Plugging these values into the formula, we get:

a_n = 19 \cdot (-6)^{n-1}

Simplifying the Formula

We can simplify the formula by expanding the exponent:

a_n = 19 \cdot (-6)^{n-1}

a_n = 19 \cdot (-6)^{n} \cdot (-6)^{-1}

a_n = 19 \cdot (-6)^{n} \cdot \frac{1}{-6}

a_n = 19 \cdot (-6)^{n} \cdot \frac{1}{6}

a_n = \frac{19 \cdot (-6)^{n}}{6}

Conclusion

In this article, we derived an explicit formula for the nth term of a geometric sequence, given by the sequence 19, 13, 7, ... . The formula is:

a_n = \frac{19 \cdot (-6)^{n}}{6}

This formula can be used to find the nth term of the sequence for any positive integer value of n.

Example Use Cases

Here are a few example use cases for the formula:

Find the 5th term of the sequence: $a_5 = \frac{19 \cdot (-6)^{5}}{6}$
Find the 10th term of the sequence: $a_{10} = \frac{19 \cdot (-6)^{10}}{6}$
Find the 15th term of the sequence: $a_{15} = \frac{19 \cdot (-6)^{15}}{6}$

Limitations

The formula derived in this article is only applicable to geometric sequences with a common ratio of -6. If the common ratio is different, a different formula will be required.

Future Work

In future work, we can explore other types of sequences and derive explicit formulas for their nth terms. We can also investigate the properties of geometric sequences and their applications in real-world problems.

References

[1] "Geometric Sequences" by Math Open Reference
[2] "Sequences and Series" by Khan Academy

Appendix

The following is a list of formulas for the nth term of a geometric sequence with different common ratios:

For a common ratio of 2: $a_n = a_1 \cdot 2^{n-1}$
For a common ratio of 3: $a_n = a_1 \cdot 3^{n-1}$
For a common ratio of -2: $a_n = a_1 \cdot (-2)^{n-1}$
For a common ratio of -3: $a_n = a_1 \cdot (-3)^{n-1}$
Q&A: Geometric Sequences and Explicit Formulas =====================================================

Introduction

In our previous article, we derived an explicit formula for the nth term of a geometric sequence, given by the sequence 19, 13, 7, ... . In this article, we will answer some frequently asked questions about geometric sequences and explicit formulas.

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, known as the common ratio.

Q: How do I find the common ratio of a geometric sequence?

A: To find the common ratio, you can divide each term by its previous term. For example, if the sequence is 2, 6, 18, ..., you can divide 6 by 2 to get a common ratio of 3.

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 \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: How do I use the formula to find the nth term of a geometric sequence?

A: To use the formula, you need to plug in the values of $a_1$ , $r$ , and $n$ . For example, if the sequence is 2, 6, 18, ... and you want to find the 5th term, you can plug in $a_1 = 2$ , $r = 3$ , and $n = 5$ into the formula:

a_5 = 2 \cdot 3^{5-1}

Q: What if the common ratio is not an integer?

A: If the common ratio is not an integer, you can still use the formula to find the nth term of the sequence. For example, if the sequence is 2, 6, 18, ... and the common ratio is 3/2, you can plug in $a_1 = 2$ , $r = 3/2$ , and $n = 5$ into the formula:

a_5 = 2 \cdot \left(\frac{3}{2}\right)^{5-1}

Q: Can I use the formula to find the sum of a geometric sequence?

A: Yes, you can use the formula to find the sum of a geometric sequence. The sum of a geometric sequence is given by the formula:

S_n = \frac{a_1 \cdot (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 if the sequence is not geometric?

A: If the sequence is not geometric, you may need to use a different formula or method to find the nth term or sum of the sequence. For example, if the sequence is 1, 4, 9, 16, ..., you can use the formula for the nth term of a quadratic sequence:

a_n = a_1 + (n-1) \cdot d

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