Writing a Rule for the nth Term

Oct 13, 2025 by pinkandtrash.com 76 views

Introduction

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. In this article, we will explore how to write a rule for the nth term of a geometric sequence and then use this rule to find the 9th term of a given sequence.

Writing a Rule for the nth Term

To write a rule for the nth term of a geometric sequence, we need to identify the first term and the common ratio. The first term is the first number in the sequence, and the common ratio is the number by which each term is multiplied to get the next term.

In the given sequence, the first term is -1, and the common ratio is -4/1 = -4. Therefore, the rule for the nth term of this sequence is:

a_n = (-1)(-4)^{n-1}

This rule can be used to find any term in the sequence by substituting the value of n into the equation.

Finding the 9th Term

Now that we have the rule for the nth term, we can use it to find the 9th term of the sequence. To do this, we substitute n = 9 into the equation:

a_9 = (-1)(-4)^{9-1}

a_9 = (-1)(-4)^8

a_9 = (-1)(65536)

a_9 = -65536

Therefore, the 9th term of the sequence is -65536.

Understanding Geometric Sequences

Geometric sequences are an important type of sequence in mathematics, and they have many real-world applications. They are used in finance to calculate compound interest, in physics to model population growth, and in engineering to design electrical circuits.

Properties of Geometric Sequences

Geometric sequences have several important properties that make them useful in mathematics and other fields. Some of these properties include:

Common Ratio: The common ratio is the number by which each term is multiplied to get the next term. In the given sequence, the common ratio is -4.
First Term: The first term is the first number in the sequence. In the given sequence, the first term is -1.
nth Term: The nth term is the term in the sequence that corresponds to the nth position. In the given sequence, the 9th term is -65536.
Sum of a Geometric Series: The sum of a geometric series is the sum of all the terms in the sequence. The formula for the sum of a geometric series is:

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

where a is the first term, r is the common ratio, and n is the number of terms.

Real-World Applications of Geometric Sequences

Geometric sequences have many real-world applications in finance, physics, engineering, and other fields. Some examples of real-world applications of geometric sequences include:

Compound Interest: Geometric sequences are used in finance to calculate compound interest. Compound interest is the interest earned on both the principal amount and any accrued interest over time.
Population Growth: Geometric sequences are used in physics to model population growth. Population growth is the increase in the number of individuals in a population over time.
Electrical Circuits: Geometric sequences are used in engineering to design electrical circuits. Electrical circuits are used to transmit and distribute electrical energy.

Conclusion

In this article, we explored how to write a rule for the nth term of a geometric sequence and then used this rule to find the 9th term of a given sequence. We also discussed the properties of geometric sequences and their real-world applications. Geometric sequences are an important type of sequence in mathematics, and they have many real-world applications in finance, physics, engineering, and other fields.

References

[1] "Geometric Sequences." Math Open Reference, mathopenref.com/sequences-geometric.html.
[2] "Geometric Sequences and Series." Khan Academy, khanacademy.org/math/sequences-series-summation/sequences-series-summation-new/sequences-series-summation-new/v/geometric-sequences-and-series.
[3] "Geometric Sequences." Wolfram MathWorld, mathworld.wolfram.com/GeometricSequence.html.
Geometric Sequences: Q&A ==========================

Q: What is a geometric sequence?

A: 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.

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

A: To find the common ratio of a geometric sequence, you can divide any term by the previous term. For example, if the sequence is -1, 4, -16, 64, ..., you can divide the second term by the first term to get the common ratio: 4 / -1 = -4.

Q: How do I write a rule for the nth term of a geometric sequence?

A: To write a rule for the nth term of a geometric sequence, you need to identify the first term and the common ratio. The rule for the nth term is given by:

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

where a1 is the first term and r is the common ratio.

Q: How do I find the sum of a geometric series?

A: The sum of a geometric series is given by the formula:

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

where a is the first term, r is the common ratio, and n is the number of terms.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: 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. An arithmetic sequence, on the other hand, is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

Q: How do I determine if a sequence is geometric or arithmetic?

A: To determine if a sequence is geometric or arithmetic, you can look at the relationship between consecutive terms. If the ratio between consecutive terms is constant, the sequence is geometric. If the difference between consecutive terms is constant, the sequence is arithmetic.

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

A: Geometric sequences have many real-world applications in finance, physics, engineering, and other fields. Some examples include:

Compound Interest: Geometric sequences are used in finance to calculate compound interest.
Population Growth: Geometric sequences are used in physics to model population growth.
Electrical Circuits: Geometric sequences are used in engineering to design electrical circuits.

Q: How do I use geometric sequences to model population growth?

A: To use geometric sequences to model population growth, you can start with an initial population and then multiply it by a growth factor to get the population at a later time. For example, if the initial population is 100 and the growth factor is 2, the population after one year would be 200, after two years would be 400, and so on.

Q: How do I use geometric sequences to design electrical circuits?

A: To use geometric sequences to design electrical circuits, you can use the formula for the sum of a geometric series to calculate the total resistance of a circuit. For example, if you have a circuit with three resistors in series, each with a resistance of 10 ohms, the total resistance would be:

R_{total} = 10 + 10 \cdot 10 + 10 \cdot 10^2

This is an example of a geometric sequence, where each term is multiplied by 10 to get the next term.

Conclusion

In this article, we answered some common questions about geometric sequences, including how to find the common ratio, write a rule for the nth term, and find the sum of a geometric series. We also discussed some real-world applications of geometric sequences and how to use them to model population growth and design electrical circuits.