Geometric Series & Sequences
Master geometric sequences, series, formulas, and their applications with interactive examples and exponential growth patterns
📈 Geometric Sequences
Definition
A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a constant value called the common ratio (r).
General Form:
$$a, ar, ar^2, ar^3, \ldots$$
Examples
2, 6, 18, 54, 162, ...
r = 3 (exponential growth)
100, 50, 25, 12.5, 6.25, ...
r = 0.5 (exponential decay)
3, -6, 12, -24, 48, ...
r = -2 (alternating signs)
5, 5, 5, 5, 5, ...
r = 1 (constant sequence)
Interactive Sequence Generator
Generated Sequence:
Pattern: Each term = Previous term ×
Behavior Analysis
Growth (|r| > 1)
Terms grow exponentially larger
r = 2: 1, 2, 4, 8, 16, ...
r = -1.5: 1, -1.5, 2.25, -3.375, ...
Decay (|r| < 1)
Terms approach zero
r = 0.5: 1, 0.5, 0.25, 0.125, ...
r = -0.3: 1, -0.3, 0.09, -0.027, ...
Special Cases
Constant or alternating
r = 1: 5, 5, 5, 5, ...
r = -1: 3, -3, 3, -3, ...
∑ Geometric Series
Definition
A geometric series is the sum of the terms in a geometric sequence. Unlike arithmetic series, geometric series can have finite sums even with infinite terms when |r| < 1.
Series Form:
$$S = a + ar + ar^2 + ar^3 + \cdots$$
Visual Example
Sequence: 3, 6, 12, 24, 48
Series: 3 + 6 + 12 + 24 + 48
Sum = 93
Series Calculator
Terms:
Sum:
Infinite Geometric Series
Convergence Condition
An infinite geometric series converges (has a finite sum) only when |r| < 1.
Infinite Sum Formula:
$$S_\infty = \frac{a}{1-r} \quad \text{when } |r| < 1$$
Interactive Calculator
📐 Essential Formulas
Sequence Formulas
General Term (nth term)
$$a_n = a \cdot r^{n-1}$$
$a_n$ = nth term
$a$ = first term
$r$ = common ratio
$n$ = term position
Common Ratio
$$r = \frac{a_{n+1}}{a_n}$$
Ratio between consecutive terms
Series Formulas
Finite Sum (r ≠ 1)
$$S_n = a \cdot \frac{1-r^n}{1-r}$$
Sum of first n terms
Infinite Sum (|r| < 1)
$$S_\infty = \frac{a}{1-r}$$
Sum of infinite terms
Special Case (r = 1)
$$S_n = n \cdot a$$
When ratio equals 1
Formula Practice Problems
Solution:
🌍 Real-World Applications
Interactive Application Problem
🦠 Bacterial Growth Problem
A bacterial culture starts with an initial population and doubles every hour. Calculate the population after several hours and the total bacteria produced.
Population Each Hour:
Final Population:
Total Bacteria Produced (Cumulative):
🧠 Geometric Sequences & Series Quiz
Quiz Complete! 🎉
Review Your Answers
Explanation:
Key Takeaways
Geometric vs Arithmetic
- • Geometric: Multiply by constant ratio
- • Arithmetic: Add constant difference
- • Geometric grows exponentially
- • Can have infinite sums when |r| < 1
Essential Formulas
- • nth term: $a_n = a \cdot r^{n-1}$
- • Finite sum: $S_n = a \cdot \frac{1-r^n}{1-r}$
- • Infinite sum: $S_\infty = \frac{a}{1-r}$ (|r| < 1)