What is an arithmetic sequence?

An arithmetic sequence has a constant difference between consecutive terms. Formula: an = a1 + (n-1)d, where d is the common difference. Example: 2, 5, 8, 11 (d = 3).

What is a geometric sequence?

A geometric sequence has a constant ratio between consecutive terms. Formula: an = a1 * r^(n-1), where r is the common ratio. Example: 3, 6, 12, 24 (r = 2).

What is the Fibonacci sequence?

Fibonacci: each term is the sum of the two preceding terms. Sequence: 1, 1, 2, 3, 5, 8, 13, 21... The ratio of consecutive terms approaches the golden ratio (phi approx 1.618).

How do I find the nth term of a sequence?

First identify the pattern: constant difference = arithmetic, constant ratio = geometric. For arithmetic: an = a1 + (n-1)d. For geometric: an = a1 * r^(n-1). For others, find the recurrence relation.

What are common applications of sequences?

Arithmetic: loan payments, salary increases. Geometric: compound interest, population growth, depreciation. Fibonacci: nature patterns, art, algorithm design.

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Number Sequence Calculator

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Sequence Calculator

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Sequence Type

First Term (a₁)

Common Difference (d)

nth Term to Find

Sequence Formulas & Properties

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, ...

Real-World Applications

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)

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Frequently Asked Questions

An arithmetic sequence has a constant difference (d) between consecutive terms. Formula: aₙ = a₁ + (n-1)d. Sum formula: Sₙ = n/2 × (2a₁ + (n-1)d) or Sₙ = n/2 × (a₁ + aₙ).

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