An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, and it characterizes the linear progression of the sequence.
In contrast, a geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Geometric sequences exhibit exponential growth or decay, showcasing a multiplicative pattern. While arithmetic sequences demonstrate a constant additive change between terms, geometric sequences highlight a consistent multiplicative relationship, leading to distinct patterns of progression in the two types of sequences.
Comparison Chart
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Definition | A sequence of numbers where the difference between consecutive terms is constant. | A sequence of numbers where the ratio between consecutive terms is constant. |
| Formula for nth term | a_n = a_1 + d(n-1) | a_n = a_1 * r^(n-1) |
| Common element | Common difference (d) | Common ratio (r) |
| Change between terms | Constant addition or subtraction | Constant multiplication or division |
| Graph | Creates a straight line | Creates an exponential curve (increases or decreases depending on the sign of r) |
| Examples | 3, 7, 11, 15, … | 2, 6, 18, 54, … (increasing) |
| Real-world applications | – Equally spaced stair steps – Counting objects with a constant increment (e.g., apples on a tree) – Uniform acceleration | – Population growth with a constant rate of increase – Compound interest calculations – Decaying radioactive materials |
| Sum of terms (series) | Sn = n/2 * (a_1 + a_n) | Sn = a_1 * (1 – r^n) / (1 – r) (when r ≠ 1) |
| Infinite series | May diverge (grow without bound) or converge (approach a specific value) depending on the common difference. | May diverge (grow or shrink without bound) or converge (approach a specific value) depending on the absolute value of the common ratio. |
What is Arithmetic Sequence?
An arithmetic sequence, also known as an arithmetic progression, is a set of numbers in which the difference between any two consecutive terms is constant. This difference, denoted as “d,” is referred to as the common difference. Arithmetic sequences play a fundamental role in mathematics and are essential in various fields, including physics, finance, and computer science.
Definition and Notation
In an arithmetic sequence, the “n”-th term (an) can be expressed as the sum of the initial term (a₁) and the product of the common difference (d) and (n-1). The general form of an arithmetic sequence is given by:
an = a₁ + (n-1)d
Here, a₁ represents the first term, d is the common difference, and n denotes the position of the term in the sequence.
Generating Terms
To generate terms in an arithmetic sequence, one can use the formula an = a₁ + (n-1)d. This allows for the determination of any term in the sequence without having to explicitly list all the preceding terms. The common difference plays a crucial role in defining the pattern and spacing between consecutive elements.
Sum of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence, referred to as the arithmetic series, can be calculated using the formula:
Sn = (n/2)[2a₁ + (n-1)d]
This formula simplifies the process of finding the sum of a specific number of terms in an arithmetic sequence, providing a concise expression for the cumulative total.
Examples and Applications
Arithmetic sequences are prevalent in real-world scenarios. For instance, financial analysts use them to model the linear growth of investments or depreciation of assets over time. In physics, these sequences describe uniform motion, where the position of an object changes by a constant amount in equal time intervals.
Examples of Arithmetic Sequences
- 2, 5, 8, 11, 14, …
- 10, 7, 4, 1, -2, …
- 3, 7, 11, 15, 19, …
- -6, -2, 2, 6, 10, …
- 100, 90, 80, 70, 60, …
What is Geometric Sequence?
A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The general form of a geometric sequence is given by a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.
Key Characteristics
- Common Ratio (r): The common ratio is a crucial element of a geometric sequence. It determines the factor by which each term is multiplied to obtain the next one. The value of r can be positive or negative, and it distinguishes the behavior of the sequence.
- First Term (a_1): a_1 is the initial term of the sequence. It serves as the starting point from which the geometric progression unfolds.
Recursive and Explicit Formulas
- Recursive Formula: The recursive formula for a geometric sequence expresses each term in terms of the preceding one. It is given by a_n = r * a_(n-1), with the initial condition a_1.
- Explicit Formula: The explicit formula provides a direct expression for any term in the sequence without referring to the previous terms. It is given by a_n = a_1 * r^(n-1).
Properties
- Behavior of r:
- If |r| > 1, the sequence is divergent, and the terms grow without bound.
- If 0 < |r| < 1, the sequence is convergent, and the terms approach zero as n increases.
- Sum of Terms (Finite Geometric Series): The sum of the first n terms of a geometric sequence, known as a geometric series, is given by the formula S_n = (a_1 * (r^n – 1)) / (r – 1) if r ≠ 1, and S_n = n * a_1 if r = 1.
Applications
Geometric sequences are prevalent in various fields, including finance, biology, physics, and computer science. They are used to model exponential growth or decay, population dynamics, radioactive decay, and the time complexity of algorithms.
Examples of Geometric Sequences
- 3, 9, 27, 81, 243, …
- 4, 12, 36, 108, 324, …
- -2, 6, -18, 54, -162, …
- 5, 15, 45, 135, 405, …
- 80, 40, 20, 10, 5, …
Difference Between Arithmetic and Geometric Sequence
- Definition:
- Arithmetic Sequence: A sequence in which the difference between consecutive terms is constant.
- Geometric Sequence: A sequence in which the ratio of any two consecutive terms is constant.
- Common Notation:
- Arithmetic Sequence: Typically denoted as “an = a1 + (n-1)d,” where an is the nth term, a1 is the first term, and d is the common difference.
- Geometric Sequence: Commonly represented as “an = a1 * r^(n-1),” where an is the nth term, a1 is the first term, and r is the common ratio.
- Common Difference/Ratio:
- Arithmetic Sequence: The constant difference (d) between consecutive terms is the same throughout the sequence.
- Geometric Sequence: The common ratio (r) between consecutive terms remains constant.
- Expression for nth Term:
- Arithmetic Sequence: “an = a1 + (n-1)d”
- Geometric Sequence: “an = a1 * r^(n-1)”
- Sum of First n Terms:
- Arithmetic Sequence: The sum of the first n terms (Sn) is given by “Sn = (n/2)[2a1 + (n-1)d].”
- Geometric Sequence: The sum of the first n terms (Sn) is given by “Sn = a1(r^n – 1)/(r-1).”
- Graphical Representation:
- Arithmetic Sequence: The graph is a straight line.
- Geometric Sequence: The graph is an exponential curve.
- Divergence/Convergence:
- Arithmetic Sequence: The terms continue to increase or decrease linearly.
- Geometric Sequence: The terms grow or shrink exponentially, possibly diverging to infinity or converging to zero.
- Applications:
- Arithmetic Sequence: Commonly used to model linear growth or arithmetic progressions.
- Geometric Sequence: Useful for modeling exponential growth or decay situations.
- Example:
- Arithmetic Sequence: 2, 5, 8, 11, 14, … with a common difference d = 3.
- Geometric Sequence: 3, 9, 27, 81, 243, … with a common ratio r = 3.
References
- Rahmani-Andebili, Mehdi. “Problems: Arithmetic and Geometric Sequences and Series.” Precalculus. Springer, Cham, 2021. 101-105.
- Ross, Amanda. “A Brief Look at Arithmetic and Geometric Sequences.” Pedagogy and Content in Middle and High School Mathematics. Brill, 2017. 77-80.