Introduction
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The nth term of a geometric sequence can be written as:
an = a1 * r^(n-1), where a1 is the first term, r is the common ratio, and n is the term number.
Characteristics of Geometric Sequences
- Constant Ratio: In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, denoted as ‘r’.
- Exponential Growth or Decay: Geometric sequences exhibit exponential growth or decay depending on whether the common ratio is greater than 1 (growth) or between 0 and 1 (decay).
- Distinct Multiples: The terms of a geometric sequence are multiples of each other, differing only by the common ratio ‘r’.
Graphical Representation of Geometric Sequences
Graphs can visually represent sequences, including geometric sequences. In the context of geometric sequences, graphs can help illustrate the growth or decay pattern of the sequence based on the common ratio ‘r’.
Types of Graphs for Geometric Sequences
1. Line Graph
A line graph is a commonly used graph type to represent geometric sequences. Each point on the graph represents a term in the geometric sequence, with the x-axis representing the term number ‘n’ and the y-axis representing the value of the term ‘an’.
For a geometric sequence with a positive common ratio ‘r’, the line graph will exhibit an exponential growth pattern, where the slope of the line increases as ‘n’ increases. Conversely, for a geometric sequence with a common ratio ‘r’ between 0 and 1, the line graph will show an exponential decay pattern, with the slope decreasing as ‘n’ increases.
2. Exponential Function Graph
An exponential function graph, such as y = a * r^x, can also represent a geometric sequence. In this case, ‘x’ represents the term number, ‘a’ is the first term of the sequence, and ‘r’ is the common ratio.
For a geometric sequence, the exponential function graph will exhibit a curve that either rises or falls exponentially based on the common ratio ‘r’. If ‘r’ is greater than 1, the graph will rise sharply, indicating growth. If ‘r’ is between 0 and 1, the graph will decline gradually, indicating decay.
Identifying the Graph of a Geometric Sequence
When presented with multiple graphs, it is important to identify which graph represents a geometric sequence based on its characteristics.
Key Characteristics to Look for in the Graph
- Exponential Growth or Decay: Look for a graph that shows either exponential growth (for r > 1) or exponential decay (for 0 < r < 1) patterns.
- Constant Ratio: Identify if the graph exhibits a consistent multiplication factor between terms, indicating a fixed common ratio ‘r’.
- Distinct Multiples: Check if the values on the graph are multiples of each other, differing by the common ratio ‘r’.
Steps to Determine the Graph Representing a Geometric Sequence
- Identify the first term ‘a1’ and the common ratio ‘r’ of the sequence.
- Plot the values of the geometric sequence on the graph based on the term number ‘n’ and the value of the term ‘an’.
- Observe the pattern of the graph to determine if it exhibits exponential growth or decay and if the values follow a consistent multiplication factor.
- Compare the graph with the characteristics of a geometric sequence to confirm if it represents a geometric sequence based on the given ‘r’ value.
Examples of Graphs Representing Geometric Sequences
Let’s consider two examples of graphs representing geometric sequences with different common ratios:
Example 1: Geometric Sequence with r = 2
For a geometric sequence with a common ratio ‘r’ of 2, the graph will exhibit exponential growth. The values of the terms will double with each subsequent term. The graph will show a steep upward curve as ‘n’ increases.
Example 2: Geometric Sequence with r = 0.5
For a geometric sequence with a common ratio ‘r’ of 0.5, the graph will display exponential decay. The values of the terms will halve with each subsequent term. The graph will show a gradual downward curve as ‘n’ increases.
Conclusion
Graphs are valuable tools for representing geometric sequences visually and understanding their growth or decay patterns. By analyzing the characteristics and patterns in the graphs, one can identify which graph represents a geometric sequence based on the common ratio and multiplicative properties.
