# Finding the Recursive Formula of a Geometric Sequence: Example 6

## Topic

Sequences and Series

## Description

Process for Finding the Recursive Formula

1. Identify the First Term: The first term of the sequence is denoted as a1.
2. Determine the Common Ratio: The common ratio r is found by dividing the second term by the first term.
3. Write the Recursive Formula: The recursive formula for an arithmetic sequence is:

an = an - 1•r

where an is the nth term, an - 1 is the previous term, and r is the common ratio.

Distinguishing Recursive from Explicit Formulas

1. Recursive Formula: Defines each term based on the previous term(s). It requires knowing the initial term and is useful for generating terms sequentially.
2. Explicit Formula: Allows direct computation of any term in the sequence without reference to previous terms. It is more efficient for finding terms far into the sequence.

Given Sequence

Sequence: [5, 25, 125, 625, 3125]

First term (a₁) = 5

Common ratio (r) = 25 / 5 = 5

Recursive formula: an = an - 1•5

For a complete collection of math examples related to Sequences and Series click on this link: Math Examples: Sequences and Series Collection.

Common Core Standards CCSS.MATH.CONTENT.HSF.BF.A.2 9 - 11 Algebra     • Sequences and Series         • Sequences 2022 geometric sequence, recursive formula