N-th Term

Definition

The n-th term of a sequence is the term that appears in the nth position of the sequence. It represents the specific value of the sequence at a given index or position, where n is a positive integer denoting the ordinal number of the term within the sequence.

5 Must Know Facts For Your Next Test

1. The n-th term formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
2. The n-th term can be used to find any term in the sequence, given the first term and the common ratio.
3. Knowing the n-th term formula allows you to predict and calculate future terms in the sequence without needing to generate the entire sequence.
4. The n-th term is a crucial concept in understanding and working with geometric sequences, as it provides a concise way to represent and manipulate the terms of the sequence.
5. Geometric sequences have many real-world applications, such as in finance, biology, and physics, where the n-th term formula is used to model and analyze various phenomena.

Review Questions

• Explain how the n-th term formula can be used to generate any term in a geometric sequence.
  • The n-th term formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. By substituting different values of $n$ into this formula, you can calculate the value of any term in the sequence. For example, if the first term is 2 and the common ratio is 3, the 5th term would be $a_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 162$. This demonstrates how the n-th term formula allows you to determine the value of a specific term without needing to generate the entire sequence.
• Describe how the n-th term concept is used to analyze and model real-world phenomena involving geometric sequences.
  • The n-th term formula for geometric sequences is widely used to model and analyze various real-world phenomena that exhibit exponential growth or decay patterns. For instance, in finance, the n-th term formula can be used to calculate the future value of an investment that earns a constant interest rate, compounded over time. In biology, the n-th term formula can be used to model population growth, where each generation is a multiple of the previous one. In physics, the n-th term formula can be used to describe the decay of radioactive materials or the propagation of sound waves. By applying the n-th term concept, researchers and professionals can make accurate predictions, analyze trends, and gain insights into the underlying patterns of these real-world processes.
• Evaluate the importance of understanding the n-th term concept in the context of geometric sequences and their applications.
  • The n-th term concept is a fundamental and indispensable tool in the study of geometric sequences and their applications. By understanding the n-th term formula, $a_n = a_1 \cdot r^{n-1}$, you can not only generate any term in the sequence but also gain a deeper understanding of the underlying patterns and properties of geometric sequences. This knowledge is crucial in a wide range of fields, from finance and economics to biology and physics, where geometric sequences are used to model and analyze real-world phenomena. The ability to apply the n-th term formula allows for accurate predictions, efficient calculations, and the development of sophisticated models that can inform decision-making and drive scientific discoveries. Mastering the n-th term concept is therefore a vital skill for success in advanced mathematics and its diverse applications.

Related terms

Sequence:

A sequence is an ordered list of elements, where each element is called a term, and the position of each term within the list is denoted by a natural number, called the index or the term number.

Geometric Sequence:

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a common ratio, which is a constant value.

Common Ratio:

The common ratio of a geometric sequence is the constant value by which each term is multiplied to obtain the next term in the sequence.