Arithmetic Sequence: (0,2), (1,4), (2,6), (3,8)
This arithmetic sequence represents a pattern where each term increases by 2. The coordinates (0,2), (1,4), (2,6), (3,8) correspond to the terms of the sequence. The first value in each pair (0, 1, 2, 3) represents the index or position of the term in the sequence, and it is plotted on the horizontal axis. The second value in each pair (2, 4, 6, 8) represents the value of the term itself, and it is plotted on the vertical axis.
The numbers are presented as integers (2, 4, 6, 8) rather than decimal numbers (2.0, 4.0, 6.0, 8.0) because the sequence follows a pattern of whole numbers. In mathematics, it is common to use the simplest form of a number, so integers are used when decimal places are not needed.
The plot below illustrates this sequence as discrete points, showing the linear relationship between the terms:
The arithmetic sequence is a fundamental concept in mathematics, often used to model linear growth or to describe patterns that follow a constant difference. In this case, the constant difference is 2, meaning that each term is 2 greater than the previous term.
Geometric Sequence: (0,1), (1,3), (2,9), (3,27)
This geometric sequence represents a pattern where each term is multiplied by 3. The coordinates (0,1), (1,3), (2,9), (3,27) correspond to the terms of the sequence. The first value in each pair (0, 1, 2, 3) represents the index or position of the term in the sequence, and it is plotted on the horizontal axis. The second value in each pair (1, 3, 9, 27) represents the value of the term itself, and it is plotted on the vertical axis.
The numbers are presented as integers because the sequence follows a pattern of whole numbers, and there are no fractional parts in these terms.
The plot below illustrates this sequence as discrete points, showing the exponential growth pattern between the terms:
The geometric sequence is a fundamental concept in mathematics, often used to model exponential growth or decay, or to describe patterns that follow a constant multiplication factor. In this case, the constant multiplication factor is 3, meaning that each term is 3 times greater than the previous term.
Fibonacci Sequence: (0,1), (1,2), (2,3), (3,5), (4,8), (5,13)
This sequence represents the famous Fibonacci pattern, where each term is the sum of the two preceding terms. The coordinates (0,1), (1,2), (2,3), (3,5), (4,8), (5,13) correspond to the terms of the sequence. The first value in each pair (0, 1, 2, 3, 4, 5) represents the index or position of the term in the sequence, and it is plotted on the horizontal axis. The second value in each pair (1, 2, 3, 5, 8, 13) represents the value of the term itself, and it is plotted on the vertical axis.
The numbers are presented as integers, reflecting the whole number nature of the Fibonacci sequence.
The plot below illustrates this sequence as discrete points, showing the growth pattern between the terms:
The Fibonacci sequence is a fundamental concept in mathematics, often found in nature and art, and used to model growth patterns and various phenomena. It follows a recursive pattern where each term is the sum of the two preceding terms.
Exponential Decay Sequence: (0,8), (1,4), (2,2), (3,1)
This sequence represents an exponential decay pattern, where each term is halved. The coordinates (0,8), (1,4), (2,2), (3,1) correspond to the terms of the sequence. The first value in each pair (0, 1, 2, 3) represents the index or position of the term in the sequence, and it is plotted on the horizontal axis. The second value in each pair (8, 4, 2, 1) represents the value of the term itself, and it is plotted on the vertical axis.
The numbers are presented as integers, reflecting the whole number nature of this sequence.
The plot below illustrates this sequence as discrete points, showing the exponential decay pattern between the terms:
The exponential decay sequence is a fundamental concept in mathematics, often used to model decay, depreciation, or reduction patterns in various contexts. It follows a pattern where each term is a fixed fraction (in this case, one-half) of the previous term.