Sequences and Series
Visualizing Arithmetic and Geometric Progressions
Explore how arithmetic and geometric sequences behave differently. Adjust the parameters to see how the sequence changes and observe the patterns in both tabular and graphical representations!
Arithmetic Sequence
Sequence Values:
1, 3, 5, 7, 9, 11
Observation:
Arithmetic sequences show a constant difference between terms, resulting in a linear pattern when graphed.
The Mathematics Behind Sequences
Key Concepts:
Sequences are ordered lists of numbers following specific patterns:
- Arithmetic Sequence: Each term increases by a constant difference (d). Formula: aₙ = a₁ + (n-1)d
- Geometric Sequence: Each term is multiplied by a constant ratio (r). Formula: aₙ = a₁ × rⁿ⁻¹
- Series: The sum of terms in a sequence
Real-world Applications:
Sequences model many real-world phenomena like loan payments (arithmetic), population growth (geometric), fractals (geometric), and more. Understanding sequences helps in financial planning, computer algorithms, and scientific modeling.
