Calculating the First Four Terms of Sequence 1

Oct 13, 2025 by windailys.com 61 views

Introduction

Sequences are an essential concept in mathematics, and they play a vital role in various mathematical disciplines, including algebra, geometry, and calculus. A sequence is a series of numbers that follow a specific pattern or rule. In this article, we will explore two different sequences defined by given rules and write the first four terms of each sequence.

Sequence 1: $f(n) = n^3 - 1$

The first sequence is defined by the rule $f(n) = n^3 - 1$ . This means that to find the nth term of the sequence, we need to substitute n into the equation and simplify. Let's calculate the first four terms of this sequence.

Calculating the First Four Terms of Sequence 1

To find the first four terms of the sequence, we need to substitute n = 1, 2, 3, and 4 into the equation $f(n) = n^3 - 1$ .

For n = 1, we have $f(1) = 1^3 - 1 = 0$ .
For n = 2, we have $f(2) = 2^3 - 1 = 7$ .
For n = 3, we have $f(3) = 3^3 - 1 = 26$ .
For n = 4, we have $f(4) = 4^3 - 1 = 63$ .

Therefore, the first four terms of the sequence defined by the rule $f(n) = n^3 - 1$ are 0, 7, 26, and 63.

Sequence 2: $f(n) = 2n^2 + 4$

The second sequence is defined by the rule $f(n) = 2n^2 + 4$ . This means that to find the nth term of the sequence, we need to substitute n into the equation and simplify. Let's calculate the first four terms of this sequence.

Calculating the First Four Terms of Sequence 2

To find the first four terms of the sequence, we need to substitute n = 1, 2, 3, and 4 into the equation $f(n) = 2n^2 + 4$ .

For n = 1, we have $f(1) = 2(1)^2 + 4 = 6$ .
For n = 2, we have $f(2) = 2(2)^2 + 4 = 16$ .
For n = 3, we have $f(3) = 2(3)^2 + 4 = 32$ .
For n = 4, we have $f(4) = 2(4)^2 + 4 = 52$ .

Therefore, the first four terms of the sequence defined by the rule $f(n) = 2n^2 + 4$ are 6, 16, 32, and 52.

Conclusion

Introduction

Sequences are an essential concept in mathematics, and they play a vital role in various mathematical disciplines, including algebra, geometry, and calculus. In our previous article, we explored two different sequences defined by given rules and wrote the first four terms of each sequence. In this article, we will answer some frequently asked questions about sequences and provide additional information to help you better understand this concept.

Q&A

Q: What is a sequence?

A: A sequence is a series of numbers that follow a specific pattern or rule. It is a way of representing a set of numbers in a particular order.

Q: How do I determine the nth term of a sequence?

A: To determine the nth term of a sequence, you need to substitute n into the equation that defines the sequence and simplify.

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

A: A sequence is a list of numbers that follow a specific pattern or rule, while a series is the sum of the terms of a sequence.

Q: How do I find the sum of a sequence?

A: To find the sum of a sequence, you need to add up all the terms of the sequence. This can be done using various methods, including arithmetic series, geometric series, and infinite series.

Q: What is the formula for the nth term of a sequence?

A: The formula for the nth term of a sequence depends on the specific sequence. For example, the formula for the nth term of the sequence $f(n) = n^3 - 1$ is $f(n) = n^3 - 1$ .

Q: How do I determine the first four terms of a sequence?

A: To determine the first four terms of a sequence, you need to substitute n = 1, 2, 3, and 4 into the equation that defines the sequence and simplify.

Q: What is the significance of sequences in real-life applications?

A: Sequences have numerous real-life applications, including finance, economics, computer science, and engineering. They are used to model population growth, predict stock prices, and design algorithms.

Q: How do I use sequences to solve problems?

A: To use sequences to solve problems, you need to identify the pattern or rule that defines the sequence and then use that information to solve the problem.

Examples of Sequences in Real-Life Applications

Example 1: Population Growth

Suppose a population grows at a rate of 2% per year. The sequence that represents the population growth can be defined as $f(n) = 100(1.02)^n$ , where n is the number of years.

Example 2: Stock Prices

Suppose a stock price increases by 5% per year. The sequence that represents the stock price can be defined as $f(n) = 100(1.05)^n$ , where n is the number of years.

Example 3: Algorithm Design

Suppose we want to design an algorithm that sorts a list of numbers. The sequence that represents the sorted list can be defined as $f(n) = n!$ , where n is the number of elements in the list.

Conclusion

In this article, we answered some frequently asked questions about sequences and provided additional information to help you better understand this concept. We also explored some examples of sequences in real-life applications, including population growth, stock prices, and algorithm design. By understanding sequences, you can better analyze and solve problems in various fields of study.