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

First Term (a)

Common Ratio (r)

Number of Terms

Generated Sequence:

2, 6, 18, 54, 162, 486

Pattern: Each term = Previous term × 3

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

First Term (a)

Common Ratio (r)

Number of Terms (n)

Terms:

4 + 8 + 16 + 32 + 64

Sum:

S₅ = 124

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

First Term (a)

Common Ratio (r)

Infinite Sum:

S∞ = 2

📐 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

Problem 1: Find the 8th term

Given sequence: 5, 15, 45, 135, ...

Solution:

Given: $a = 5$ , $r = 3$

Using: $a_n = a \cdot r^{n-1}$

$a_8 = 5 \cdot 3^{8-1} = 5 \cdot 3^7$

$a_8 = 5 \cdot 2187 = 10935$

$a_8 = 10,935$

Problem 2: Find infinite sum

Series: 12 + 6 + 3 + 1.5 + ...

Solution:

Given: $a = 12$ , $r = 0.5$

Since $|r| = 0.5 < 1$ , series converges

Using: $S_\infty = \frac{a}{1-r}$

$S_\infty = \frac{12}{1-0.5} = \frac{12}{0.5}$

$S_\infty = 24$

Problem 3: Sum of first 6 terms

Series: 2 + 8 + 32 + 128 + ...

Solution:

Given: $a = 2$ , $r = 4$ , $n = 6$

Using: $S_n = a \cdot \frac{1-r^n}{1-r}$

$S_6 = 2 \cdot \frac{1-4^6}{1-4} = 2 \cdot \frac{1-4096}{-3}$

$S_6 = 2 \cdot \frac{-4095}{-3} = 2 \cdot 1365$

$S_6 = 2,730$

Problem 4: Find the ratio

If $a_3 = 18$ and $a_6 = 486$ , find r

Solution:

$a_3 = a \cdot r^2 = 18$

$a_6 = a \cdot r^5 = 486$

Dividing: $\frac{a_6}{a_3} = \frac{r^5}{r^2} = r^3$

$r^3 = \frac{486}{18} = 27$

$r = \sqrt[3]{27} = 3$

🌍 Real-World Applications

🦠

Population Growth

Modeling exponential population growth in biology, bacteria cultures, or viral spread.

Example: Bacteria doubling every hour - Hour 1: 100, Hour 2: 200, Hour 3: 400...

💰

Compound Interest

Calculating investment growth, loan interest, or savings account returns over time.

Example:

1000 at 5% annual interest - Year 1:

1050, Year 2: $1102.50...

☢️

Radioactive Decay

Modeling half-life decay in physics, carbon dating, or medical isotopes.

Example: Half-life of 10 years - 100g → 50g → 25g → 12.5g...

💻

Technology Growth

Moore's Law, processing power doubling, or network effects in technology adoption.

Example: Processing power doubling - 2020: 1x, 2022: 2x, 2024: 4x...

🔺

Fractals & Geometry

Self-similar patterns, Sierpinski triangle, or geometric scaling in nature.

Example: Fractal area scaling - Level 1: 1, Level 2: 1/3, Level 3: 1/9...

📊

Economic Models

Inflation rates, economic growth models, or market expansion patterns.

Example: 3% annual inflation -

100 →

103 →

106.09 →

109.27...

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.

Initial Population

Growth Factor

Number of Hours

Population Each Hour:

50, 100, 200, 400, 800, 1600

Final Population:

1,600 bacteria

Total Bacteria Produced (Cumulative):

3,150 total bacteria

🧠 Geometric Sequences & Series Quiz

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What is the common ratio in the sequence: 3, 6, 12, 24, 48?

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Review Your Answers

Question 1

Incorrect

What is the common ratio in the sequence: 3, 6, 12, 24, 48?

A.2✓

B.3

C.4

D.6

Explanation: Each term is multiplied by 2 to get the next term: 3×2=6, 6×2=12, etc.

Question 2

Incorrect

Find the 6th term of the geometric sequence with first term a = 2 and common ratio r = 3.

A.162

B.243

C.486✓

D.729

Explanation: Using $a_n = a \cdot r^{n-1}$ : $a_6 = 2 \cdot 3^{6-1} = 2 \cdot 3^5 = 2 \cdot 243 = 486$ .

Question 3

Incorrect

Which condition must be satisfied for an infinite geometric series to converge?

A.|r| > 1

B.|r| < 1✓

C.r = 1

D.r > 0

Explanation: An infinite geometric series converges only when the absolute value of the common ratio is less than 1.

Question 4

Incorrect

Find the sum of the infinite geometric series: 8 + 4 + 2 + 1 + ...

A.15

B.16✓

C.12

D.∞

Explanation: Here a = 8 and r = 1/2. Since |r| < 1, the sum is $S_\infty = \frac{a}{1-r} = \frac{8}{1-0.5} = \frac{8}{0.5} = 16$ .

Question 5

Incorrect

In a geometric sequence, if the 3rd term is 12 and the 5th term is 48, what is the common ratio?

A.2✓

B.4

C.1.5

D.3

Explanation: If $a_3 = 12$ and $a_5 = 48$ , then $\frac{a_5}{a_3} = r^2 = \frac{48}{12} = 4$ . Therefore $r = \sqrt{4} = 2$ .

Question 6

Incorrect

What is the sum of the first 5 terms of the geometric series: 2 + 6 + 18 + 54 + ...?

A.242✓

B.162

C.324

D.486

Explanation: Here a = 2, r = 3, n = 5. Using $S_n = a \cdot \frac{1-r^n}{1-r} = 2 \cdot \frac{1-3^5}{1-3} = 2 \cdot \frac{1-243}{-2} = 2 \cdot \frac{-242}{-2} = 242$ .

Question 7

Incorrect

A ball is dropped from 100 feet and bounces to 3/4 of its previous height each time. What is the total distance traveled?

A.700 feet✓

B.400 feet

C.600 feet

D.500 feet

Explanation: The ball falls 100 feet, then bounces up 75 feet and falls 75 feet, bounces up 56.25 feet and falls 56.25 feet, etc. Total = 100 + 2(75 + 56.25 + ...) = 100 + 2 × 75/(1-0.75) = 100 + 2 × 300 = 700 feet.

Question 8

Incorrect

Which of the following is NOT a geometric sequence?

A.1, 4, 16, 64, 256

B.5, 15, 45, 135, 405

C.2, 6, 18, 54, 162

D.3, 6, 12, 20, 30✓

Explanation: In option D: 6/3 = 2, 12/6 = 2, but 20/12 = 1.67 and 30/20 = 1.5. The ratio is not constant, so it's not geometric.

Question 9

Incorrect

If the first term of a geometric sequence is 7 and the common ratio is -2, what is the 4th term?

A.-56✓

B.56

C.-28

D.28

Explanation: Using $a_n = a \cdot r^{n-1}$ : $a_4 = 7 \cdot (-2)^{4-1} = 7 \cdot (-2)^3 = 7 \cdot (-8) = -56$ .

Question 10

Incorrect

The sum of an infinite geometric series is 12 and the first term is 4. What is the common ratio?

A.1/3

B.2/3✓

C.1/2

D.3/4

Explanation: Using $S_\infty = \frac{a}{1-r}$ : $12 = \frac{4}{1-r}$ . Solving: $12(1-r) = 4$ , so $12 - 12r = 4$ , thus $12r = 8$ and $r = \frac{2}{3}$ .

Geometric Series & Sequences

📈 Geometric Sequences

Definition

Examples

Interactive Sequence Generator

Generated Sequence:

Behavior Analysis

Growth (|r| > 1)

Decay (|r| < 1)

Special Cases

∑ Geometric Series

Definition

Visual Example

Series Calculator

Terms:

Sum:

Infinite Geometric Series

Convergence Condition

Interactive Calculator

📐 Essential Formulas

Sequence Formulas

General Term (nth term)

Common Ratio

Series Formulas

Finite Sum (r ≠ 1)

Infinite Sum (|r| < 1)

Special Case (r = 1)

Formula Practice Problems

Problem 1: Find the 8th term

Problem 2: Find infinite sum

Problem 3: Sum of first 6 terms

Problem 4: Find the ratio

🌍 Real-World Applications

Population Growth

Compound Interest

Radioactive Decay

Technology Growth

Fractals & Geometry

Economic Models

Interactive Application Problem

🦠 Bacterial Growth Problem

Population Each Hour:

Final Population:

Total Bacteria Produced (Cumulative):

🧠 Geometric Sequences & Series Quiz

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Key Takeaways

Geometric vs Arithmetic

Essential Formulas