Recursive Formula

Definition

A recursive formula is a mathematical expression that defines a sequence or series by relating each term to the previous term(s) in the sequence. It allows for the generation of a sequence by repeatedly applying the same rule or formula to generate the next term based on the preceding term(s).

5 Must Know Facts For Your Next Test

1. Recursive formulas are commonly used to describe and generate sequences, such as arithmetic sequences and geometric sequences.
2. The recursive formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
3. The recursive 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.
4. Recursive formulas can be used to define series, such as the sum of an arithmetic or geometric sequence, by relating each term to the previous term(s).
5. Recursive formulas often require an initial or starting value, known as the seed value, to begin the iterative process of generating the sequence.

Review Questions

• Explain how a recursive formula differs from an explicit formula in the context of sequences.
  • The key difference between a recursive formula and an explicit formula for a sequence is the way they define each term. A recursive formula defines each term in the sequence based on the previous term(s), using a repeating rule or relationship. In contrast, an explicit formula directly expresses each term as a function of its position or index within the sequence, without relying on the previous terms. Recursive formulas are more flexible and can be used to generate a wider range of sequences, while explicit formulas provide a more direct and compact way to represent certain types of sequences, such as arithmetic and geometric sequences.
• Describe how a recursive formula can be used to generate an arithmetic sequence and a geometric sequence.
  • For an arithmetic sequence, the recursive formula is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference. This formula allows you to generate each term by adding the common difference to the previous term. For a geometric sequence, the recursive formula is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. This formula generates each term by multiplying the previous term by the common ratio. The recursive nature of these formulas enables the iterative process of building up the sequence term by term, starting from the initial or seed value.
• Explain how recursive formulas can be used to define and calculate the sum of an arithmetic or geometric series.
  • Recursive formulas can be used to define the terms of a series, which is the sum of the terms in a sequence. For an arithmetic series, the recursive formula for the $n$th term can be used to express the sum of the first $n$ terms as a function of the first term, common difference, and the number of terms. Similarly, for a geometric series, the recursive formula for the $n$th term can be used to derive a formula for the sum of the first $n$ terms, which depends on the first term, common ratio, and the number of terms. These recursive relationships allow for the efficient calculation of series sums, which are important in various mathematical and scientific applications.

Related terms

Sequence:

A sequence is an ordered list of numbers, objects, or other mathematical entities that follow a specific pattern or rule.

Explicit Formula:

An explicit formula is a mathematical expression that directly defines each term of a sequence or series in terms of its position or index within the sequence.

Iteration: Iteration is the process of repeatedly applying the same rule or formula to generate the next term in a sequence or series.