How Do You Write a Geometric Recursive Rule?

A geometric recursive rule defines a term in a sequence by relating it to the preceding term, specifically by multiplying the previous term by a constant ratio.

Here's how to write a geometric recursive rule:

1. Identify the First Term: Determine the first term in the sequence, denoted as a₁. This is your starting point.

2. Determine the Common Ratio: Calculate the common ratio, denoted as r. This is the factor you multiply by to get from one term to the next. You can find r by dividing any term by its preceding term (e.g., a₂ / a₁, a₃ / a₂, etc.).

3. Write the Recursive Formula: Express the nth term, a_n, in terms of the (n-1)th term, a_n-1, and the common ratio, r. The general form of a geometric recursive rule is:

   • a₁ = (the value of the first term)
   • a_n = a_n-1 r , for n* > 1

Example

Let's say you have the geometric sequence: 2, 6, 18, 54, ...

1. First Term: a₁ = 2

2. Common Ratio: r = 6 / 2 = 3 (or 18 / 6 = 3, or 54 / 18 = 3)

3. Recursive Rule:

   • a₁ = 2
   • a_n = a_n-1 3 , for n* > 1

This rule states that the first term is 2, and to find any subsequent term, you multiply the previous term by 3.

Explanation

The recursive rule breaks down the sequence into a step-by-step process. You start with the initial value (a₁), and then the second part of the rule tells you how to get the next value based on the previous one. This is different from an explicit formula, which allows you to calculate any term directly without knowing the preceding terms.