Describe the graphs of an arithmetic and a geometric sequence. Compare the graphs to a linear and an exponential function.
The Correct answer and Explanation is:
Description of Arithmetic and Geometric Sequence Graphs
- Arithmetic Sequence:
- An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. For example, in the sequence 2, 4, 6, 8, …, the common difference is 2.
- Graph: The graph of an arithmetic sequence forms a series of discrete points along a straight line with a constant slope. It resembles a linear function but is distinct because only specific integer values of nnn are plotted (typically starting at n=1n = 1n=1). The graph is discrete and not continuous, meaning there are gaps between points.
- Geometric Sequence:
- A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a constant ratio. For example, in the sequence 2, 4, 8, 16, …, the common ratio is 2.
- Graph: The graph of a geometric sequence forms a series of discrete points that follow an exponential curve, as the values grow (or shrink) rapidly. The points are still discrete and separated, but they curve upwards (for positive growth) or downwards (for decay).
Comparison to Linear and Exponential Functions
- Linear Function: A linear function has the general form y=mx+by = mx + by=mx+b and represents a continuous straight line where the rate of change is constant. The graph of an arithmetic sequence, which consists of equally spaced points along a straight line, mirrors a linear function. However, the arithmetic sequence is not continuous like the linear function; it only has values at integer points.
- Exponential Function: An exponential function has the form y=abxy = ab^xy=abx (with b>0b > 0b>0) and represents a continuous curve where the rate of change accelerates or decelerates exponentially. The graph of a geometric sequence mirrors an exponential function because the terms of the geometric sequence grow (or decay) in a multiplicative fashion. However, like the arithmetic sequence, the geometric sequence is discrete, with points separated on the graph rather than forming a continuous curve.
In summary, arithmetic and geometric sequences graph similarly to linear and exponential functions but differ in being discrete sets of points rather than continuous lines or curves.