R^n

Definition

The term $r^n$ represents the exponential expression where $r$ is the base and $n$ is the exponent. This expression is fundamental in the context of geometric sequences, as it describes the relationship between consecutive terms in the sequence.

5 Must Know Facts For Your Next Test

1. In a geometric sequence, the $n$-th term is given by the formula $a_n = a_1 extbackslash cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
2. The common ratio $r$ is the base of the exponential expression $r^n$ that describes the relationship between consecutive terms in a geometric sequence.
3. The value of $r^n$ determines the rate of growth or decay in a geometric sequence, with $r > 1$ indicating growth and $0 < r < 1$ indicating decay.
4. Geometric sequences are often used to model real-world phenomena that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest.
5. The expression $r^n$ is crucial in deriving formulas for the $n$-th term, the sum of the first $n$ terms, and other important properties of geometric sequences.

Review Questions

• Explain how the term $r^n$ is used to describe the relationship between consecutive terms in a geometric sequence.
  • In a geometric sequence, the ratio between consecutive terms is constant and represented by the variable $r$. The $n$-th term of the sequence is given by the formula $a_n = a_1 extbackslash cdot r^{n-1}$, where $a_1$ is the first term. The exponential expression $r^n$ captures the multiplicative relationship between the $n$-th term and the first term, with the exponent $n-1$ accounting for the position of the term within the sequence. This expression is fundamental in understanding the pattern and behavior of geometric sequences.
• Discuss the significance of the value of $r$ in the expression $r^n$ and its impact on the growth or decay of a geometric sequence.
  • The value of the common ratio $r$ in the expression $r^n$ determines whether the geometric sequence exhibits growth or decay. If $r > 1$, the sequence will exhibit exponential growth, with each term being larger than the previous one. Conversely, if $0 < r < 1$, the sequence will exhibit exponential decay, with each term being smaller than the previous one. The rate of growth or decay is directly proportional to the value of $r$, as it determines the multiplier applied to the previous term to obtain the next term in the sequence. Understanding the behavior of $r^n$ is crucial in analyzing and predicting the long-term behavior of geometric sequences.
• Explain how the expression $r^n$ is used to derive important formulas and properties related to geometric sequences.
  • The expression $r^n$ is central to the formulas and properties of geometric sequences. The formula for the $n$-th term, $a_n = a_1 extbackslash cdot r^{n-1}$, directly incorporates $r^n$ to describe the relationship between the $n$-th term and the first term. Additionally, the formula for the sum of the first $n$ terms of a geometric sequence, $S_n = a_1 extbackslash cdot rac{1 - r^n}{1 - r}$, relies on the properties of $r^n$ to derive this expression. Understanding the behavior and manipulation of $r^n$ is essential in applying these formulas and analyzing the characteristics of geometric sequences, such as their convergence, divergence, and the identification of the common ratio.

Related terms

Geometric Sequence:

A sequence where the ratio between consecutive terms is constant, often denoted as $r$.

Common Ratio:

The constant ratio between consecutive terms in a geometric sequence, represented by the variable $r$.

Exponential Function:

A function of the form $f(x) = a extbackslash cdot b^x$, where $a$ is the initial value and $b$ is the base of the exponential.