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# Geometric Sequences Overview

Created by
@SucceedingHexagon

### Write the explicit formula for a geometric sequence.

a_n = a_1 * r^(n-1)

### Write the recursive formula for a geometric sequence.

a_n = r * a_(n-1)

### What is the common ratio (r) in a geometric sequence?

The common ratio is a number that is multiplied to the previous number to get the next number in the sequence.

### Find the common ratio of the following geometric sequence: 6, -3, (3/2), -(3/4),...

<p>r = -(1/2)</p> Signup and view all the answers

### Find the 7th term for the sequence 2, 6, 18, 54,...

<p>1458</p> Signup and view all the answers

### Find the 8th term for the sequence 3, 9, 27.

<p>6561</p> Signup and view all the answers

### Find the common ratio for a = 5 * (3)^(n-1).

<p>r = 3</p> Signup and view all the answers

### Write the explicit formula for the sequence 1, -(1/2), (1/4), -(1/8),....

<p>a_n = 1 * (-1/2)^(n-1)</p> Signup and view all the answers

### Write the explicit formula for the sequence 2, 6, 18, 54....

<p>a_n = 2 * (3)^(n-1)</p> Signup and view all the answers

### The graph of a geometric sequence is always an?

<p>Exponential curve</p> Signup and view all the answers

### The explicit formula is used to find the?

<p>nth term or any term</p> Signup and view all the answers

### The recursive formula may be used to find the?

<p>next term in the geometric sequence</p> Signup and view all the answers

## Study Notes

### Geometric Sequences Overview

• Explicit formula for a geometric sequence is used to calculate the nth term directly.
• Recursive formula for a geometric sequence calculates the next term based on the previous term.

### Common Ratio

• The common ratio (r) is the factor by which each term is multiplied to obtain the next term in the sequence.
• Example provided: For the sequence 6, -3, (3/2), -(3/4), the common ratio is r = -(1/2).

### Terms Calculation

• To find specific terms in a geometric sequence, use the formulas derived from the initial term and the common ratio.
• For the sequence 2, 6, 18, 54, the 7th term is 1458.
• For the sequence 3, 9, 27, the 8th term is 6561.

### Formulas

• The common ratio for a given formula like a = 5*(3)^(n-1) is r = 3.
• Explicit formula for the sequence 1, -(1/2), (1/4), -(1/8) is a = 1*(-1/2)^(n-1).
• Explicit formula for the sequence 2, 6, 18, 54 is a = 2*(3)^(n-1).

### Graphical Representation

• The graph of a geometric sequence is characterized by an exponential curve, illustrating rapid growth or decay.

### Purpose of Formulas

• The explicit formula is utilized to determine the nth term or to find any specific term in a geometric sequence.
• The recursive formula is advantageous for calculating the next term based on the most recent term in the sequence.

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## Description

This quiz covers geometric sequences, including both explicit and recursive formulas. You'll learn about the common ratio, how to calculate specific terms, and graphical representations of geometric sequences. Prepare to apply these concepts to various examples and improve your understanding of geometric progressions.

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