Arithmetic Sequence Calculator
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Understanding Arithmetic Sequences
Basic Concepts
An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
nth Term: aₙ = a₁ + (n-1)d
Sum of n Terms: Sₙ = n/2[2a₁ + (n-1)d]
Common Difference: d = aₙ₊₁ - aₙ
Where:
a₁ = First term
d = Common difference
n = Number of terms
Properties and Characteristics
Key Properties
- Constant difference between terms
- Linear growth pattern
- Arithmetic mean property
- Terms form a linear function
- Symmetric sum property
Special Cases
- Zero common difference
- Negative common difference
- Alternating sequences
- Integer sequences
- Finite vs. infinite sequences
Advanced Topics
Applications
- Linear Functions
- Financial Mathematics
- Data Analysis
- Number Theory
- Pattern Recognition
- Probability Sequences
Mathematical Analysis
Arithmetic Mean: AM = (aₘ + aₙ)/2
Term Position: n = (aₙ - a₁)/d + 1
Partial Sums: Sₖ = k(a₁ + aₖ)/2
Related Concepts
- Geometric Sequences
- Harmonic Sequences
- Fibonacci Sequence
- Recursive Sequences
- Series Convergence