R^n

Definition

The term $r^n$ represents the exponential function, where $r$ is the base and $n$ is the exponent. This expression is fundamental in understanding geometric sequences and series, as it describes the pattern of growth or decay in these mathematical structures.

5 Must Know Facts For Your Next Test

1. The expression $r^n$ represents the $n$-th power of the base $r$, where $n$ is a non-negative integer.
2. In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio $r$, which is represented by $r^n$.
3. The sum of a geometric series can be calculated using the formula $S = a / (1 - r)$, where $a$ is the first term and $r$ is the common ratio.
4. The value of $r$ determines whether the geometric sequence or series is increasing ($r > 1$), decreasing ($0 < r < 1$), or constant ($r = 1$).
5. The exponent $n$ in $r^n$ represents the position of the term within the geometric sequence, with the first term being $r^0$.

Review Questions

• Explain how the term $r^n$ is used to describe the pattern of a geometric sequence.
  • In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio, $r$. The term $r^n$ represents the $n$-th term in the sequence, where $n$ is the position of the term. For example, if the first term of a geometric sequence is $a$ and the common ratio is $r$, then the $n$-th term can be expressed as $a ullet r^{n-1}$, which is equivalent to $a ullet r^n$. This pattern of $r^n$ allows us to easily generate and understand the terms of a geometric sequence.
• Describe the role of $r^n$ in the formula for the sum of a geometric series.
  • The formula for the sum of a geometric series is $S = a / (1 - r)$, where $a$ is the first term and $r$ is the common ratio. The term $r^n$ is not directly present in this formula, but it is an essential component in understanding the derivation of the formula. The sum of a geometric series can be expressed as the sum of the terms $a, a ullet r, a ullet r^2, a ullet r^3, ext{dots}, a ullet r^n$. By recognizing this pattern and using the formula for the sum of a geometric series, we can efficiently calculate the total sum of the series.
• Analyze how the value of $r$ in $r^n$ affects the behavior of a geometric sequence or series.
  • The value of $r$ in the expression $r^n$ has a significant impact on the behavior of a geometric sequence or series. If $r > 1$, the sequence or series is increasing, as each term is larger than the previous one. If $0 < r < 1$, the sequence or series is decreasing, as each term is smaller than the previous one. If $r = 1$, the sequence or series is constant, as each term is equal to the previous one. Understanding the relationship between $r$ and the behavior of the sequence or series is crucial in analyzing and interpreting the patterns and trends observed in geometric sequences and series.

Related terms

Geometric Sequence: A sequence where each term is obtained by multiplying the previous term by a common ratio, $r$.

Geometric Series: The sum of the terms in a geometric sequence, where the sum is calculated using the formula $S = a / (1 - r)$, where $a$ is the first term and $r$ is the common ratio.

Common Ratio: The constant ratio between consecutive terms in a geometric sequence, denoted as $r$.