The $n^{\text{th}}$ Term Of A Sequence Is Given By $20 - 3n$.Work Out The First Three Terms In The Sequence. Give Your Answers In Order.

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Introduction

In mathematics, a sequence is a list of numbers in a specific order. The $n^{\text{th}}$ term of a sequence is the term that appears at the $n^{\text{th}}$ position in the sequence. In this article, we will explore how to calculate the first three terms of a sequence given by the formula $20 - 3n$.

Understanding the Formula

The formula $20 - 3n$ represents the $n^{\text{th}}$ term of the sequence. To calculate the first three terms, we need to substitute $n$ with the values 1, 2, and 3, respectively.

Substituting $n$ with 1

When $n = 1$, the formula becomes:

$$20 - 3(1) = 20 - 3 = 17$$

So, the first term of the sequence is 17.

Substituting $n$ with 2

When $n = 2$, the formula becomes:

$$20 - 3(2) = 20 - 6 = 14$$

So, the second term of the sequence is 14.

Substituting $n$ with 3

When $n = 3$, the formula becomes:

$$20 - 3(3) = 20 - 9 = 11$$

So, the third term of the sequence is 11.

Conclusion

In this article, we have calculated the first three terms of a sequence given by the formula $20 - 3n$. By substituting $n$ with the values 1, 2, and 3, we found that the first three terms of the sequence are 17, 14, and 11, respectively. This demonstrates how to use the formula to calculate the $n^{\text{th}}$ term of a sequence.

The First Three Terms of the Sequence

Term	Value
1	17
2	14
3	11

Why is this Important?

Understanding how to calculate the $n^{\text{th}}$ term of a sequence is crucial in mathematics, particularly in algebra and calculus. It allows us to analyze and solve problems involving sequences and series, which have numerous applications in physics, engineering, and economics.

Real-World Applications

Sequences and series have many real-world applications, including:

• Physics: Sequences and series are used to model the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
• Engineering: Sequences and series are used to design and analyze electrical circuits, mechanical systems, and other complex systems.
• Economics: Sequences and series are used to model economic systems, such as the behavior of stock prices or the growth of a population.

Conclusion

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Introduction

In our previous article, we explored how to calculate the first three terms of a sequence given by the formula $20 - 3n$. In this article, we will answer some frequently asked questions (FAQs) about sequences and series.

Q&A

Q: What is a sequence?

A: A sequence is a list of numbers in a specific order. Each number in the list is called a term, and the position of the term is called the index or subscript.

Q: What is the $n^{\text{th}}$ term of a sequence?

A: The $n^{\text{th}}$ term of a sequence is the term that appears at the $n^{\text{th}}$ position in the sequence. It is denoted by the formula $a_n$, where $a$ is the general term of the sequence.

Q: How do I calculate the $n^{\text{th}}$ term of a sequence?

A: To calculate the $n^{\text{th}}$ term of a sequence, you need to substitute $n$ with the desired value in the formula $a_n$. For example, if you want to calculate the 5th term of a sequence, you would substitute $n$ with 5 in the formula.

Q: What is the difference between a sequence and a series?

A: A sequence is a list of numbers in a specific order, while a series is the sum of the terms of a sequence. For example, the sequence $1, 2, 3, 4, 5$ has a sum of $15$, which is a series.

Q: How do I determine if a sequence is convergent or divergent?

A: A sequence is convergent if it approaches a finite limit as $n$ approaches infinity. A sequence is divergent if it does not approach a finite limit as $n$ approaches infinity. To determine if a sequence is convergent or divergent, you can use various tests, such as the ratio test or the root test.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence 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.

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

A: To find the sum of an infinite geometric series, you need to use the formula $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. The series converges only if $|r| < 1$.

Conclusion

In conclusion, sequences and series are fundamental concepts in mathematics that have numerous applications in physics, engineering, and economics. By understanding how to calculate the $n^{\text{th}}$ term of a sequence and answering frequently asked questions, we can analyze and solve problems involving sequences and series.

Common Mistakes to Avoid

• Not substituting $n$ with the correct value: Make sure to substitute $n$ with the correct value in the formula $a_n$.
• Not checking for convergence: Make sure to check if the sequence is convergent or divergent before calculating the sum.
• Not using the correct formula: Make sure to use the correct formula for the sum of a geometric sequence or an infinite geometric series.

Real-World Applications

Sequences and series have many real-world applications, including:

• Physics: Sequences and series are used to model the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
• Engineering: Sequences and series are used to design and analyze electrical circuits, mechanical systems, and other complex systems.
• Economics: Sequences and series are used to model economic systems, such as the behavior of stock prices or the growth of a population.

Conclusion

In conclusion, sequences and series are fundamental concepts in mathematics that have numerous applications in physics, engineering, and economics. By understanding how to calculate the $n^{\text{th}}$ term of a sequence and answering frequently asked questions, we can analyze and solve problems involving sequences and series.