Explicit Formulas

Definition

Explicit formulas, in the context of geometric sequences, refer to a formula that directly expresses the nth term of the sequence in terms of the initial term and common ratio. This type of formula allows for the easy calculation of any term in the sequence without needing to recursively apply the common ratio.

5 Must Know Facts For Your Next Test

1. The explicit 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. Explicit formulas allow for the direct calculation of any term in the sequence without the need to recursively apply the common ratio.
3. Explicit formulas are particularly useful when working with long geometric sequences or when needing to find a specific term that is not the first few terms.
4. The common ratio, $r$, is a key component of the explicit formula and determines the rate of growth or decay in the sequence.
5. Explicit formulas can be used to analyze and make predictions about the behavior of geometric sequences, such as their growth or decay patterns.

Review Questions

• Explain how the explicit formula for a geometric sequence is derived and how it differs from a recursive formula.
  • The explicit formula for a geometric sequence, $a_n = a_1 \cdot r^{n-1}$, is derived by recognizing the pattern in the sequence where each term is obtained by multiplying the previous term by the common ratio, $r$. This formula allows you to directly calculate the nth term without needing to recursively apply the common ratio, as would be the case with a recursive formula. The explicit formula is more convenient for working with long sequences or finding specific terms, while the recursive formula is better suited for generating the sequence term-by-term.
• Describe how the common ratio, $r$, affects the behavior of a geometric sequence expressed using the explicit formula.
  • The common ratio, $r$, is a crucial component of the explicit formula for a geometric sequence, $a_n = a_1 \cdot r^{n-1}$. The value of $r$ determines whether the sequence is growing or decaying over time. If $r > 1$, the sequence is growing exponentially, with each term being larger than the previous one. If $0 < r < 1$, the sequence is decaying exponentially, with each term being smaller than the previous one. The magnitude of $r$ also affects the rate of growth or decay, with values of $r$ further from 1 resulting in more rapid changes in the sequence.
• Analyze how the explicit formula for a geometric sequence can be used to make predictions about the long-term behavior of the sequence.
  • The explicit formula for a geometric sequence, $a_n = a_1 \cdot r^{n-1}$, allows for the easy calculation of any term in the sequence, which can be used to make predictions about its long-term behavior. By examining the value of the common ratio, $r$, one can determine whether the sequence is growing or decaying over time. If $r > 1$, the sequence will grow exponentially, with the terms becoming larger and larger as $n$ increases. Conversely, if $0 < r < 1$, the sequence will decay exponentially, with the terms becoming smaller and smaller. The explicit formula also enables the analysis of the rate of growth or decay, which can be used to forecast the future behavior of the sequence and make informed decisions based on its long-term trajectory.

Related terms

Geometric Sequence:

A sequence where each term is obtained by multiplying the previous term by a constant ratio, known as the common ratio.

Common Ratio:

The constant ratio between consecutive terms in a geometric sequence.

Recursive Formula:

A formula that expresses each term in a sequence in terms of the previous term(s).