Geometric Sequence Calculator

Calculate geometric sequences, series sums, and explore exponential patterns

Calculate Geometric Sequence

Sequence Results

a₁ = 2
First Term
r = 3
Common Ratio
162
a(5)

General formula: aₙ = a₁ × r^(n-1) = 2 × 3^(n-1)

Explicit formula: aₙ = 2 × 3^(n-1)

Recursive formula: aₙ = aₙ₋₁ × 3, a₁ = 2

Finite Series Sum

to
S = 242

Infinite Series Sum

Divergent (|r| ≥ 1)
S∞ = Divergent

Geometric Sequence Terms

a1
2
a2
6
a3
18
a4
54
a5
162
a6
486
a7
1,458
a8
4,374
a9
13,122
a10
39,366

Pattern: Each term = previous term × 3

Formula: aₙ = 2 × 3^(n-1)

Sequence Properties

Type:Increasing
Behavior:Same sign
Convergence:Divergent
Growth Rate:Exponential

Common Ratios

r = 21, 2, 4, 8, 16...
r = 0.51, 0.5, 0.25, 0.125...
r = -11, -1, 1, -1...
r = 31, 3, 9, 27, 81...
r = 0.11, 0.1, 0.01, 0.001...

Key Formulas

Nth Term
aₙ = a₁ × r^(n-1)
Finite Sum
S = a₁(r^n - 1)/(r - 1)
Infinite Sum
S = a₁/(1 - r), |r| < 1
Common Ratio
r = aₙ₊₁/aₙ

Understanding Geometric Sequences

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

Key Components

  • First term (a₁): The starting value of the sequence
  • Common ratio (r): The multiplier between consecutive terms
  • Nth term (aₙ): Any term in the sequence at position n

Explicit Formula

aₙ = a₁ × r^(n-1)

where n is the position of the term

Geometric Series

A geometric series is the sum of terms in a geometric sequence. The sum depends on whether the series is finite or infinite, and whether it converges.

Finite Series Sum

S = a₁ × (r^n - 1) / (r - 1)

for r ≠ 1, where n is the number of terms

Infinite Series Sum

S∞ = a₁ / (1 - r)

only when |r| < 1 (convergent series)

Real-World Applications

  • 💰Compound interest calculations
  • 📈Population growth models
  • 🔬Radioactive decay
  • 💻Computer memory and binary systems