Number Sequence Calculator

Arithmetic Sequence:

• nth term: $a_n = a_1 + (n-1)d$
• Sum: $S_n = \frac{n(2a_1 + (n-1)d)}{2}$
• Property: Each term increases by constant difference d
• Example: 2, 5, 8, 11, ... (d = 3)

Geometric Sequence:

• nth term: $a_n = a_1 \cdot r^{n-1}$
• Sum: $S_n = a_1 \frac{1-r^n}{1-r}$ (r ≠ 1)
• Property: Each term multiplied by constant ratio r
• Example: 3, 6, 12, 24, ... (r = 2)

Fibonacci Sequence:

• Recurrence: $F_n = F_{n-1} + F_{n-2}$
• Binet's formula: $F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$
• Property: Sum of two preceding terms
• Example: 0, 1, 1, 2, 3, 5, 8, 13, ...

Arithmetic Sequences

• • Linear growth patterns (saving money monthly)
• • Temperature changes over time
• • Seating arrangements in theaters
• • Loan payments with equal installments

Geometric Sequences

• • Population growth models
• • Compound interest calculations
• • Radioactive decay processes
• • Computer memory sizes (powers of 2)

Fibonacci Sequence

• • Flower petal arrangements in nature
• • Spiral patterns (sunflower seeds, shells)
• • Financial market analysis (Fibonacci retracements)
• • Algorithm optimization (Fibonacci search)