What is the Difference Between Arithmetic and Geometric Series?
🆚 Go to Comparative Table 🆚The main difference between arithmetic and geometric sequences lies in the relationship between their consecutive terms.
Arithmetic Sequence:
- In an arithmetic sequence, there is a constant difference between each consecutive pair of terms.
- The difference between any two consecutive terms is always the same, and it can be positive, negative, or zero.
- The general formula for an arithmetic sequence is: $$a, a + d, a + 2d, a + 3d, a + 4d, \dots$$, where $$a$$ is the first term and $$d$$ is the common difference between terms.
- Examples of arithmetic sequences include: $$2,4,6,8,10,12,14,16,18,20$$ or $$5,11,17,23,29,35,41,47,53,67,73$$.
Geometric Sequence:
- In a geometric sequence, there is a constant ratio between each consecutive pair of terms.
- The ratio between any two consecutive terms is always the same, and it can be greater than 1, less than 1, or equal to 1.
- The general formula for a geometric sequence is: $$a, ar, ar^2, ar^3, ar^4, \dots$$, where $$a$$ is the first term and $$r$$ is the common ratio.
- Examples of geometric sequences include: $$2,4,8,16,32,64,128,256,512,1024$$ or $$5,15,45,135,405,1215$$.
In summary, arithmetic sequences have a constant difference between consecutive terms, while geometric sequences have a constant ratio between consecutive terms. The variation in the elements of an arithmetic sequence is linear, while the variation in the elements of a geometric sequence is exponential.
On this pageWhat is the Difference Between Arithmetic and Geometric Series? Comparative Table: Arithmetic vs Geometric Series
Comparative Table: Arithmetic vs Geometric Series
Here is a table highlighting the differences between arithmetic and geometric series:
Feature | Arithmetic Series | Geometric Series |
---|---|---|
Definition | A sequence of integers in which the difference between consecutive terms is constant. | A sequence of numbers in which the ratio of consecutive terms is constant. |
Common Difference/Ratio | Each consecutive pair of terms has a constant difference. | Each consecutive pair of terms has a constant ratio. |
General Term | $$a_n = a + (n - 1)d$$, where $$a$$ is the first term and $$d$$ is the common difference. | $$a_n = a \cdot r^{n-1}$$, where $$a$$ is the first term and $$r$$ is the common ratio. |
Examples | 5, 11, 17, 23, 29, 35, … (with a common difference of 6). | 2, 6, 18, 54, … (with a common ratio of 3). |
In an arithmetic series, the difference between consecutive terms is always the same, while in a geometric series, the ratio between consecutive terms is always the same.
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- Numerical Expression vs Algebraic Expression
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- Algebra vs Calculus
- Mathematics vs Applied Mathematics
- Geometry vs Trigonometry
- Numeracy vs Mathematics
- Algebra vs Trigonometry
- Numeric vs Decimal
- Linear Equation vs Quadratic Equation
- Logarithmic vs Exponential
- Difference Equation vs Differential Equation
- Algebraic Expressions vs Equations
- Pattern vs Sequence