Find $s(16$\] For The Sequence $s(n) = 4n - 2$, Where $n \in \{1, 2, 3, \ldots\}$.A. 64 B. 60 C. 66 D. 62

Introduction

In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers in a specific order, and each term is determined by a rule or formula. In this article, we will explore the sequence $s(n) = 4n - 2$, where $n$ is a positive integer. Our goal is to find the 16th term of this sequence, denoted as $s(16)$.

Understanding the Sequence

The given sequence is defined by the formula $s(n) = 4n - 2$. This means that each term of the sequence is obtained by multiplying the term number $n$ by 4 and then subtracting 2. For example, the first term $s(1)$ is calculated as $4(1) - 2 = 2$, the second term $s(2)$ is calculated as $4(2) - 2 = 6$, and so on.

Calculating the 16th Term

To find the 16th term $s(16)$, we simply substitute $n = 16$ into the formula $s(n) = 4n - 2$. This gives us:

$s(16) = 4(16) - 2$

To evaluate this expression, we first multiply 4 by 16, which gives us 64. Then, we subtract 2 from 64, resulting in:

$s(16) = 64 - 2 = 62$

Conclusion

In this article, we have found the 16th term of the sequence $s(n) = 4n - 2$ by substituting $n = 16$ into the formula. The result is $s(16) = 62$. This demonstrates the importance of understanding sequences and their formulas in mathematics.

Discussion

The sequence $s(n) = 4n - 2$ is a simple yet powerful example of a linear sequence. By understanding the formula and applying it to find the 16th term, we have demonstrated the value of mathematical reasoning and problem-solving skills.

Key Takeaways

• Sequences are an essential concept in mathematics that help us understand patterns and relationships between numbers.
• The formula $s(n) = 4n - 2$ defines a linear sequence where each term is obtained by multiplying the term number $n$ by 4 and then subtracting 2.
• To find the 16th term $s(16)$, we substitute $n = 16$ into the formula and evaluate the expression.

Further Exploration

For further exploration, we can try finding other terms of the sequence $s(n) = 4n - 2$ by substituting different values of $n$ into the formula. We can also investigate other types of sequences, such as quadratic or exponential sequences, and explore their properties and applications.

References

• [1] "Sequences and Series" by Michael Sullivan, Pearson Education, 2013.
• [2] "Mathematics for the Nonmathematician" by Morris Kline, Dover Publications, 1985.

Glossary

• Sequence: A list of numbers in a specific order, where each term is determined by a rule or formula.
• Term: A single number in a sequence, identified by its position in the sequence.
• Formula: A mathematical expression that defines a sequence and determines each term.

Related Topics

• Linear Sequences
• Quadratic Sequences
• Exponential Sequences

Linear Sequences

A linear sequence is a sequence where each term is obtained by adding a fixed constant to the previous term. The formula for a linear sequence is given by:

$s(n) = an + b$

where $a$ and $b$ are constants.

Quadratic Sequences

A quadratic sequence is a sequence where each term is obtained by squaring the term number $n$ and adding a fixed constant. The formula for a quadratic sequence is given by:

$s(n) = an^2 + b$

where $a$ and $b$ are constants.

Exponential Sequences

An exponential sequence is a sequence where each term is obtained by multiplying the previous term by a fixed constant. The formula for an exponential sequence is given by:

$s(n) = ar^n$

where $a$ and $r$ are constants.