Arithmetic Sequences

What is a Sequence?

A sequence is an ordered list of numbers called terms.

Example: 2, 4, 6, 8, 10, ...

Notation:

• a₁ = first term
• a₂ = second term
• a₃ = third term
• aₙ = nth term

Types of sequences:

• Arithmetic (constant difference)
• Geometric (constant ratio)
• Other patterns

What is an Arithmetic Sequence?

An arithmetic sequence has a constant difference between consecutive terms.

Examples:

• 3, 7, 11, 15, 19, ... (add 4 each time)
• 20, 15, 10, 5, 0, ... (subtract 5 each time)
• 1, 1.5, 2, 2.5, 3, ... (add 0.5 each time)

The constant difference is called the common difference (d).

Finding the Common Difference

The common difference (d) is found by subtracting consecutive terms.

Formula: d = a₂ - a₁ = a₃ - a₂ = aₙ₊₁ - aₙ

Example 1: 5, 9, 13, 17, 21

d = 9 - 5 = 4 Check: 13 - 9 = 4 ✓ Common difference: d = 4

Example 2: 30, 25, 20, 15, 10

d = 25 - 30 = -5 Common difference: d = -5 (decreasing sequence)

Example 3: 2, 5, 8, 11, 14

d = 5 - 2 = 3 Common difference: d = 3

Example 4: 7, 7, 7, 7, 7

d = 7 - 7 = 0 Common difference: d = 0 (constant sequence)

Writing the Next Terms

Add the common difference to find the next term.

Example 1: Find next 3 terms of 4, 7, 10, 13, ...

d = 3 Next terms: 13 + 3 = 16 16 + 3 = 19 19 + 3 = 22

Sequence: 4, 7, 10, 13, 16, 19, 22, ...

Example 2: Find next 3 terms of 100, 95, 90, 85, ...

d = -5 Next terms: 85 - 5 = 80 80 - 5 = 75 75 - 5 = 70

Sequence: 100, 95, 90, 85, 80, 75, 70, ...

The Explicit Formula

The explicit formula (or nth term formula) lets you find any term directly.

Formula: aₙ = a₁ + (n - 1)d

where:

• aₙ = nth term
• a₁ = first term
• n = position number
• d = common difference

Example 1: Find the 20th term of 3, 7, 11, 15, ...

a₁ = 3, d = 4, n = 20

a₂₀ = 3 + (20 - 1)(4) a₂₀ = 3 + 19(4) a₂₀ = 3 + 76 a₂₀ = 79

Example 2: Find the 50th term of 100, 95, 90, ...

a₁ = 100, d = -5, n = 50

a₅₀ = 100 + (50 - 1)(-5) a₅₀ = 100 + 49(-5) a₅₀ = 100 - 245 a₅₀ = -145

Example 3: Find the 15th term of 2, 9, 16, 23, ...

a₁ = 2, d = 7

a₁₅ = 2 + (15 - 1)(7) a₁₅ = 2 + 98 a₁₅ = 100

Writing the Explicit Formula

Given the sequence, write the formula for aₙ.

Example 1: 5, 8, 11, 14, 17, ...

a₁ = 5, d = 3

Formula: aₙ = 5 + (n - 1)(3) Simplified: aₙ = 5 + 3n - 3 = 3n + 2

Example 2: 20, 17, 14, 11, 8, ...

a₁ = 20, d = -3

Formula: aₙ = 20 + (n - 1)(-3) Simplified: aₙ = 20 - 3n + 3 = -3n + 23

Example 3: 4, 9, 14, 19, 24, ...

a₁ = 4, d = 5

Formula: aₙ = 4 + (n - 1)(5) = 5n - 1

Finding Which Term Has a Given Value

Set aₙ equal to the value and solve for n.

Example 1: In the sequence 3, 7, 11, 15, ..., which term equals 99?

Formula: aₙ = 3 + (n - 1)(4) = 4n - 1

Set equal: 4n - 1 = 99 4n = 100 n = 25

The 25th term equals 99.

Example 2: In sequence 50, 46, 42, 38, ..., which term equals 2?

Formula: aₙ = 50 + (n - 1)(-4) = -4n + 54

-4n + 54 = 2 -4n = -52 n = 13

The 13th term equals 2.

Example 3: In sequence 1, 6, 11, 16, ..., which term equals 101?

aₙ = 1 + (n - 1)(5) = 5n - 4

5n - 4 = 101 5n = 105 n = 21

The 21st term is 101.

Recursive Formula

A recursive formula defines each term using the previous term.

General form: aₙ = aₙ₋₁ + d with a₁ given

Example 1: 5, 9, 13, 17, ...

a₁ = 5 aₙ = aₙ₋₁ + 4

Example 2: 30, 25, 20, 15, ...

a₁ = 30 aₙ = aₙ₋₁ - 5

Recursive vs. Explicit:

• Recursive: need previous term
• Explicit: can find any term directly

Graphing Arithmetic Sequences

Plot points (n, aₙ) where n is term number.

Example: 2, 5, 8, 11, 14, ...

Points: (1, 2), (2, 5), (3, 8), (4, 11), (5, 14)

Observation: Points lie on a straight line! Slope of line = common difference d

The graph is discrete (individual points), not continuous.

Arithmetic Means

An arithmetic mean between two numbers is the average.

Example: Find arithmetic mean between 10 and 20.

Mean = (10 + 20)/2 = 15

Multiple means: Insert numbers to form arithmetic sequence.

Example: Insert 3 arithmetic means between 5 and 25.

We need: 5, ?, ?, ?, 25 (5 terms total)

a₁ = 5, a₅ = 25, n = 5

Use formula: 25 = 5 + (5 - 1)d 25 = 5 + 4d 20 = 4d d = 5

Sequence: 5, 10, 15, 20, 25

Sum of Arithmetic Sequence

The sum of n terms (Sₙ) has a special formula.

Formula 1: Sₙ = n(a₁ + aₙ)/2

Formula 2: Sₙ = n[2a₁ + (n - 1)d]/2

Example 1: Find sum of first 10 terms of 3, 7, 11, 15, ...

First find a₁₀: a₁₀ = 3 + (10 - 1)(4) = 39

Use Formula 1: S₁₀ = 10(3 + 39)/2 S₁₀ = 10(42)/2 S₁₀ = 210

Example 2: Find sum: 2 + 5 + 8 + ... (20 terms)

a₁ = 2, d = 3, n = 20

Use Formula 2: S₂₀ = 20[2(2) + (20 - 1)(3)]/2 S₂₀ = 20[4 + 57]/2 S₂₀ = 20(61)/2 S₂₀ = 610

Example 3: Sum of 1 + 2 + 3 + ... + 100

a₁ = 1, aₙ = 100, n = 100

S₁₀₀ = 100(1 + 100)/2 S₁₀₀ = 100(101)/2 S₁₀₀ = 5,050

Why the Sum Formula Works

Visual explanation: Sum = 1 + 2 + 3 + 4 + 5 Reverse = 5 + 4 + 3 + 2 + 1 Add: 6 + 6 + 6 + 6 + 6 = 5 × 6 = 30 So sum = 30/2 = 15

General: Sₙ = n(first + last)/2

Applications: Saving Money

Example: Save $10 first week,$15 second week, $20 third week, etc. How much saved after 20 weeks?

a₁ = 10, d = 5, n = 20

First find a₂₀: a₂₀ = 10 + (20 - 1)(5) = 105

Sum: S₂₀ = 20(10 + 105)/2 S₂₀ = 20(115)/2 S₂₀ = $1,150

Applications: Seating

Example: Theater has 20 rows. First row has 15 seats, each row has 2 more seats than previous. How many total seats?

a₁ = 15, d = 2, n = 20

a₂₀ = 15 + 19(2) = 53 seats in last row

Total: S₂₀ = 20(15 + 53)/2 = 20(68)/2 = 680 seats

Applications: Falling Objects

Example: Object falls 16 ft in 1st second, 48 ft in 2nd second, 80 ft in 3rd second, etc. How far in 10 seconds?

Sequence: 16, 48, 80, ... d = 32

a₁₀ = 16 + 9(32) = 304 ft in 10th second

Total distance: S₁₀ = 10(16 + 304)/2 = 1,600 ft

Identifying Arithmetic Sequences

Check if sequence is arithmetic: Calculate differences between consecutive terms. If all equal, it's arithmetic!

Example 1: 3, 6, 12, 24, ... 6 - 3 = 3 12 - 6 = 6 (different!) NOT arithmetic (this is geometric)

Example 2: 5, 8, 11, 14, ... 8 - 5 = 3 11 - 8 = 3 14 - 11 = 3 ✓ Arithmetic with d = 3

Common Mistakes to Avoid

1. Confusing term number with term value a₅ = 20 means 5th term is 20, not "a times 5"

2. Wrong formula Use aₙ = a₁ + (n - 1)d, not aₙ = a₁ + nd

3. Arithmetic errors with negatives If d = -3, then add -3 (subtract 3)

4. Forgetting to find d first Always identify common difference!

5. Not checking if sequence is arithmetic Calculate differences to verify

6. Mixing up sum formulas Sₙ needs (a₁ + aₙ)/2, not (a₁ + d)

Arithmetic vs. Geometric

Arithmetic: constant difference (add/subtract)

• 2, 5, 8, 11, ... (add 3)

Geometric: constant ratio (multiply/divide)

• 2, 6, 18, 54, ... (multiply by 3)

Other patterns:

• 1, 4, 9, 16, ... (perfect squares)
• 1, 1, 2, 3, 5, 8, ... (Fibonacci)

Real-World Examples

Arithmetic sequences appear in:

• Stacking objects (height increases by constant)
• Straight-line depreciation (value decreases by constant)
• Regular savings plans
• Seating arrangements
• Odometer readings at constant speed
• Rental fees (base + per day)

Quick Reference

Common difference: d = a₂ - a₁

Explicit formula: aₙ = a₁ + (n - 1)d

Recursive formula: aₙ = aₙ₋₁ + d

Sum formula: Sₙ = n(a₁ + aₙ)/2

Practice Strategy

Level 1: Find common difference

• Given sequence, find d

Level 2: Find next terms

• Continue the pattern

Level 3: Find specific term

• Use explicit formula

Level 4: Write formula

• Given sequence, find aₙ

Level 5: Applications

• Word problems with sequences

Tips for Success

• Always find d first
• Memorize: aₙ = a₁ + (n - 1)d
• Check work by calculating a few terms
• Verify sequence is arithmetic before using formulas
• Draw diagrams for word problems
• Remember n is position number, not the term value
• Practice identifying patterns
• Use sum formula for long sequences
• Graph sequences to visualize
• Connect to linear functions (same slope as d)