Sequence & Series.
Explore arithmetic and geometric progressions, calculate sums, and visualize number patterns with interactive charts.
Sequence Type
Parameters
Sequence Visualization
Terms as Bars
Step-by-Step Solution
See exactly how the sequence is generated and summed
Understanding the Calculation
An arithmetic sequence adds a constant difference to each term.
Step 1: First term a₁ = 2, common difference d = 3
Step 2: nth term: aₙ = a₁ + (n-1) × d
Step 3: Sum: Sₙ = n/2 × (a₁ + aₙ)
Result: a₁₀ = 29, S₁₀ = 155
Smart Insights
Discover useful facts about number sequences
Arithmetic vs Geometric
Arithmetic sequences grow by adding a constant (like savings plans). Geometric sequences grow by multiplying (like compound interest).
Growth Comparison
A geometric sequence with r > 1 grows much faster than arithmetic. After 30 terms, 2 + 3n = 89, but 2 × 1.1^n ≈ 35.
Famous Sequences
The Fibonacci sequence (1, 1, 2, 3, 5, 8...) appears in nature: sunflower spirals, shell curves, and leaf arrangements.
Real-World Applications
Arithmetic sequences model salary increases and savings plans. Geometric sequences model population growth, viral spread, and compound interest.
Understanding Sequences
What is a Sequence?
A sequence is an ordered list of numbers following a specific pattern. Each number is called a term. Sequences can be finite (limited terms) or infinite (continuing forever).
Arithmetic Sequences
- Pattern: Each term differs by a constant (d)
- Formula: aₙ = a₁ + (n-1) × d
- Sum: Sₙ = n/2 × (a₁ + aₙ)
- Example: 2, 5, 8, 11, 14... (d = 3)
Geometric Sequences
- Pattern: Each term is multiplied by a constant (r)
- Formula: aₙ = a₁ × r^(n-1)
- Sum: Sₙ = a₁ × (rⁿ - 1) / (r - 1)
- Example: 3, 6, 12, 24, 48... (r = 2)
Series vs Sequences
A series is the sum of all terms in a sequence. For the sequence 2, 4, 6, 8, the series is 2 + 4 + 6 + 8 = 20. Sigma notation (Σ) is used to represent series compactly.
Convergence
An infinite geometric series converges when |r| < 1. For example: 1 + 1/2 + 1/4 + 1/8 + ... = 2. When |r| ≥ 1, the series diverges (sum grows infinitely).
Real-World Applications
- Finance: Compound interest is a geometric sequence
- Physics: Radioactive decay follows geometric decay
- Computer Science: Binary search divides by 2 each step
- Biology: Bacterial growth is exponential (geometric)
Frequently Asked Questions
What's the difference between arithmetic and geometric?
Arithmetic adds a constant difference (2, 5, 8, 11). Geometric multiplies by a constant ratio (2, 6, 18, 54). Arithmetic grows linearly; geometric grows exponentially.
How do I find the common difference?
Subtract any term from the next term: d = a₂ - a₁ = a₃ - a₂ = ... For geometric sequences, divide: r = a₂ / a₁ = a₃ / a₂.
What is the Fibonacci sequence?
Each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21... It appears in nature (sunflowers, shells) and has the golden ratio (≈1.618) as its growth factor.
Can a geometric sequence decrease?
Yes! If the common ratio r is between 0 and 1 (e.g., 0.5), the terms get smaller: 100, 50, 25, 12.5... This is called geometric decay, like radioactive half-life.
What is an infinite series?
An infinite series adds infinitely many terms. If |r| < 1 in a geometric series, it converges to a finite sum: S = a₁/(1-r). For example, 1 + 0.5 + 0.25 + ... = 2.
Where are sequences used in real life?
Sequences model salary schedules, loan payments, population growth, radioactive decay, computer algorithm efficiency (Big-O notation), and financial projections.