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📏honors pre-calculus review

key term - A_{n+1} = a_n * r

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Definition

The term $a_{n+1} = a_n * r$ is a mathematical expression that describes the relationship between consecutive terms in a geometric sequence. It represents the formula for generating the next term in the sequence, where $a_n$ is the current term and $r$ is the common ratio, or the constant multiplier between each successive term.

5 Must Know Facts For Your Next Test

  1. The term $a_{n+1} = a_n * r$ is the recursive formula for a geometric sequence, which allows you to generate the next term in the sequence given the current term and the common ratio.
  2. The common ratio $r$ is a constant value that represents the multiplier between each successive term in the geometric sequence.
  3. The recursive formula $a_{n+1} = a_n * r$ can be used to generate any term in the sequence, starting from the first term $a_1$ and the common ratio $r$.
  4. The explicit formula for a geometric sequence, $a_n = a_1 * r^{n-1}$, can be derived from the recursive formula $a_{n+1} = a_n * r$.
  5. The recursive formula $a_{n+1} = a_n * r$ is particularly useful when the first term $a_1$ and the common ratio $r$ are known, as it allows you to efficiently calculate any term in the sequence.

Review Questions

  • Explain how the recursive formula $a_{n+1} = a_n * r$ is used to generate a geometric sequence.
    • The recursive formula $a_{n+1} = a_n * r$ is used to generate a geometric sequence by starting with the first term $a_1$ and repeatedly applying the formula to calculate the next term. The common ratio $r$ is the constant multiplier that is applied to the current term $a_n$ to obtain the next term $a_{n+1}$. This allows you to generate the entire sequence by iteratively applying the recursive formula, without the need to know or remember the explicit formula for the $n$-th term.
  • Describe the relationship between the recursive formula $a_{n+1} = a_n * r$ and the explicit formula $a_n = a_1 * r^{n-1}$ for a geometric sequence.
    • The recursive formula $a_{n+1} = a_n * r$ and the explicit formula $a_n = a_1 * r^{n-1}$ for a geometric sequence are closely related. The recursive formula allows you to generate the next term in the sequence by multiplying the current term $a_n$ by the common ratio $r$. The explicit formula, on the other hand, expresses the $n$-th term directly in terms of the first term $a_1$ and the common ratio $r$. The recursive formula is the foundation for the explicit formula, as the repeated application of the recursive formula leads to the derivation of the explicit formula for a geometric sequence.
  • Analyze how the value of the common ratio $r$ affects the behavior of a geometric sequence described by the recursive formula $a_{n+1} = a_n * r$.
    • The value of the common ratio $r$ in the recursive formula $a_{n+1} = a_n * r$ has a significant impact on the behavior of the resulting geometric sequence. If $r > 1$, the sequence is an increasing geometric sequence, where each term is larger than the previous one. If $r < 1$, the sequence is a decreasing geometric sequence, where each term is smaller than the previous one. When $r = 1$, the sequence becomes an arithmetic sequence, where the common difference between terms is constant. The value of $r$ also determines the rate of growth or decay in the sequence, with larger values of $r$ leading to faster growth or decay. Understanding the relationship between the common ratio and the sequence behavior is crucial in analyzing and applying geometric sequences.