An arithmetic sequence and a geometric sequence differ in how their terms are generated. An arithmetic sequence increases or decreases by a constant difference, while a geometric sequence increases or decreases by a constant ratio. This distinction is fundamental to understanding the behavior of these two types of sequences.
Defining Arithmetic and Geometric Sequences
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| How Terms Formed | Each term is obtained by adding a constant value (common difference) to the previous term. | Each term is obtained by multiplying the previous term by a constant value (common ratio). |
| Key Operation | Addition/Subtraction | Multiplication/Division |
| Formula | a_n = a_1 + (n - 1)d (where d is the common difference) |
a_n = a_1 * r^(n-1) (where r is the common ratio) |
| Reference | "If the sequence has a common difference, it is arithmetic..." | "...if it has a common ratio, it is geometric" |
Understanding Common Difference and Common Ratio
- Common Difference (d): In an arithmetic sequence, the difference between any two consecutive terms is always the same. For example, in the sequence 2, 5, 8, 11..., the common difference is 3 (5-2=3, 8-5=3, and so on).
- Common Ratio (r): In a geometric sequence, the ratio between any two consecutive terms is always the same. For example, in the sequence 2, 4, 8, 16..., the common ratio is 2 (4/2=2, 8/4=2, and so on).
Examples to Illustrate the Differences
-
Arithmetic Sequence Example:
- Sequence: 7, 12, 17, 22, 27,...
- Common difference (d) = 5
- Each term is formed by adding 5 to the preceding term.
-
Geometric Sequence Example:
- Sequence: 3, 6, 12, 24, 48,...
- Common ratio (r) = 2
- Each term is formed by multiplying the preceding term by 2.
How to Identify Sequence Type
As the reference states, "We can therefore determine whether a sequence is arithmetic or geometric by working out whether adjacent terms differ by a common difference, or a common ratio."
Here is a step by step guide:
- Calculate Differences: Subtract each term from the next term. If you get the same value for all subtractions, then it's arithmetic.
- Calculate Ratios: Divide each term by the previous term. If you get the same value for all divisions, then it's geometric.
- If neither: If neither step gives a consistent value, then the sequence is neither arithmetic nor geometric.
Applications
Both arithmetic and geometric sequences have widespread applications:
- Arithmetic Sequences: Used in simple interest calculations, linear growth patterns, and determining even spacing.
- Geometric Sequences: Used in compound interest calculations, population growth models, and radioactive decay.
In conclusion, the key distinction between arithmetic and geometric sequences lies in their generating principles: arithmetic sequences utilize a common difference, while geometric sequences utilize a common ratio between consecutive terms.
