Write a recursive formula for the sequence: { – 9, – 27, – 81, – 243, – 729, ...} - - -

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Writing a Recursive Formula for a Sequence**

To find the recursive formula for the sequence: 

\[
\{-9, -27, -81, -243, -729, \ldots\}
\]

---

### Recursive Formula Components

1. **Initial Term:**
   - \( a_1 = \)
   - The first term of the sequence is \(-9\).

2. **Recursive Step:**
   - \( a_n = \)
   - Identify the pattern or rule that defines each subsequent term after the initial term.

### Pattern Analysis

- Observe the sequence: each term is multiplied by the same factor to obtain the next term. 
  - \( \frac{-27}{-9} = 3 \)
  - \( \frac{-81}{-27} = 3 \)
  - \( \frac{-243}{-81} = 3 \)
  - \( \frac{-729}{-243} = 3 \)
  
- This is a geometric sequence where each term is obtained by multiplying the previous term by \(-3\).

### Recursive Formula

- **Initial Term:**
  \[
  a_1 = -9
  \]

- **Recursive Rule:**
  \[
  a_n = -3 \times a_{n-1} \quad \text{for } n \geq 2
  \]

This setup allows calculation of any term in the sequence using the previous term.
Transcribed Image Text:**Writing a Recursive Formula for a Sequence** To find the recursive formula for the sequence: \[ \{-9, -27, -81, -243, -729, \ldots\} \] --- ### Recursive Formula Components 1. **Initial Term:** - \( a_1 = \) - The first term of the sequence is \(-9\). 2. **Recursive Step:** - \( a_n = \) - Identify the pattern or rule that defines each subsequent term after the initial term. ### Pattern Analysis - Observe the sequence: each term is multiplied by the same factor to obtain the next term. - \( \frac{-27}{-9} = 3 \) - \( \frac{-81}{-27} = 3 \) - \( \frac{-243}{-81} = 3 \) - \( \frac{-729}{-243} = 3 \) - This is a geometric sequence where each term is obtained by multiplying the previous term by \(-3\). ### Recursive Formula - **Initial Term:** \[ a_1 = -9 \] - **Recursive Rule:** \[ a_n = -3 \times a_{n-1} \quad \text{for } n \geq 2 \] This setup allows calculation of any term in the sequence using the previous term.
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