Learning Media: Arithmetic Series & Sequences

📊 Arithmetic Sequences

Definition

An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant value called the common difference (d).

General Form:

$$a, a+d, a+2d, a+3d, \ldots$$

Examples

2, 5, 8, 11, 14, ...

d = 3 (increasing)

20, 15, 10, 5, 0, ...

d = -5 (decreasing)

7, 7, 7, 7, 7, ...

d = 0 (constant)

Interactive Sequence Generator

First Term (a₁)

Common Difference (d)

Number of Terms

Generated Sequence:

3, 7, 11, 15, 19, 23, 27, 31

∑ Arithmetic Series

Definition

An arithmetic series is the sum of the terms in an arithmetic sequence. It represents the total when you add up all terms from the first to the nth term.

Series Form:

$$S = a_1 + a_2 + a_3 + \cdots + a_n$$

Visual Example

Sequence: 2, 5, 8, 11, 14

Series: 2 + 5 + 8 + 11 + 14

Sum = 40

Series Calculator

First Term (a₁)

Common Difference (d)

Number of Terms (n)

Terms:

5 + 8 + 11 + 14 + 17 + 20

Sum:

S₆ = 75

📐 Essential Formulas

Sequence Formulas

General Term (nth term)

$$a_n = a_1 + (n - 1)d$$

$a_n$ = nth term

$a_1$ = first term

$n$ = term position

$d$ = common difference

Common Difference

$$d = a_{n+1} - a_n$$

Difference between consecutive terms

Series Formulas

Sum of n Terms (Method 1)

$$S_n = \frac{n}{2} [2a_1 + (n-1)d]$$

Using first term and common difference

Sum of n Terms (Method 2)

$$S_n = \frac{n}{2} (a_1 + a_n)$$

Using first and last terms

Formula Practice Problems

Problem 1: Find the 15th term

Given sequence: 4, 9, 14, 19, ...

Problem 2: Find the sum

Sum of first 12 terms: 6, 11, 16, 21, ...

🌍 Real-World Applications

💰

Finance & Savings

Calculating loan payments, savings growth, or investment returns with regular contributions.

Example: Saving $50 more each month
Month 1: $100, Month 2: $150, Month 3: $200...

🏗️

Construction & Architecture

Calculating materials needed, stair dimensions, or seating arrangements in theaters.

Example: Theater seating
Row 1: 20 seats, Row 2: 22 seats, Row 3: 24 seats...

⚡

Physics & Motion

Analyzing uniform acceleration, velocity changes, or distance calculations.

Example: Free fall distances
1st sec: 5m, 2nd sec: 15m, 3rd sec: 25m...

📈

Business & Economics

Production planning, cost analysis, or revenue projections with linear growth.

Example: Production increase
Week 1: 100 units, Week 2: 125 units, Week 3: 150 units...

⏰

Time & Scheduling

Event planning, work schedules, or time management with regular intervals.

Example: Meeting schedule
9:00 AM, 9:30 AM, 10:00 AM, 10:30 AM...

👥

Population & Demographics

Analyzing linear population growth, demographic trends, or survey data.

Example: City population growth
Year 1: 50,000, Year 2: 52,000, Year 3: 54,000...

Interactive Application Problem

🏦 Savings Account Problem

Sarah starts with \$500 in her savings account and deposits an additional amount each month. If she increases her monthly deposit by \$25 each month, calculate her account balance after several months.

Initial Amount ($)

First Month Deposit ($)

Monthly Increase ($)

Number of Months

Monthly Deposits:

$100, $125, $150, $175, $200, $225

Final Balance:

$1,475

Arithmetic Series & Sequences

📊 Arithmetic Sequences

Definition

Examples

Interactive Sequence Generator

Generated Sequence:

∑ Arithmetic Series

Definition

Visual Example

Series Calculator

Terms:

Sum:

📐 Essential Formulas

Sequence Formulas

General Term (nth term)

Common Difference

Series Formulas

Sum of n Terms (Method 1)

Sum of n Terms (Method 2)

Formula Practice Problems

Problem 1: Find the 15th term

Problem 2: Find the sum

🌍 Real-World Applications

Finance & Savings

Construction & Architecture

Physics & Motion

Business & Economics

Time & Scheduling

Population & Demographics

Interactive Application Problem

🏦 Savings Account Problem

Monthly Deposits:

Final Balance:

Key Takeaways

Sequences vs Series

Essential Formulas