Sequences

Sequences: The Magic of Patterns 🎯

Imagine standing at a train station. A train arrives every 5 minutes. You know exactly when the next one comes—that’s the power of sequences!

What is a Sequence?

A sequence is just a list of numbers that follow a pattern. Like stepping stones across a river—each stone leads to the next in a predictable way.

Simple Example:

• 2, 4, 6, 8, 10… (adding 2 each time)
• 1, 2, 4, 8, 16… (doubling each time)

Think of it like this: If someone gives you the first few numbers and a rule, you can figure out ANY number in the list!

Arithmetic Sequences: The Steady Walker

What’s an Arithmetic Sequence?

Imagine you’re climbing stairs. Each step is the same height. That’s arithmetic!

An arithmetic sequence adds (or subtracts) the same number each time.

Examples:

• 3, 7, 11, 15, 19… (adding 4)
• 20, 17, 14, 11, 8… (subtracting 3)

Start → +4 → +4 → +4 → +4
  3  →  7  → 11 → 15 → 19

The Common Difference (d)

The common difference is just the “jump” between numbers.

How to find it: Subtract any term from the next one!

Example: In 5, 8, 11, 14…

• 8 - 5 = 3
• 11 - 8 = 3
• 14 - 11 = 3

The common difference d = 3

graph TD
    A["5"] -->|+3| B["8"]
    B -->|+3| C["11"]
    C -->|+3| D["14"]
    D -->|+3| E["17"]

The nth Term Formula (Arithmetic)

Want to find the 100th number without counting all 100? Use the magic formula!

Formula: aₙ = a₁ + (n - 1) × d

Where:

• aₙ = the term you want
• a₁ = first term
• n = which term (1st, 2nd, 100th…)
• d = common difference

Example: Find the 10th term of 2, 5, 8, 11…

• a₁ = 2 (first term)
• d = 3 (common difference)
• n = 10

Solution:

a₁₀ = 2 + (10 - 1) × 3
a₁₀ = 2 + 9 × 3
a₁₀ = 2 + 27
a₁₀ = 29 ✓

Geometric Sequences: The Multiplier

What’s a Geometric Sequence?

Instead of adding, we multiply by the same number each time. Like a chain reaction!

Examples:

• 2, 6, 18, 54… (×3 each time)
• 100, 50, 25, 12.5… (×0.5 each time)

Start → ×3 → ×3  → ×3  → ×3
  2  →  6 → 18  → 54  → 162

The Common Ratio ®

The common ratio is the multiplier between terms.

How to find it: Divide any term by the one before it!

Example: In 4, 12, 36, 108…

• 12 ÷ 4 = 3
• 36 ÷ 12 = 3
• 108 ÷ 36 = 3

The common ratio r = 3

graph TD
    A["4"] -->|×3| B["12"]
    B -->|×3| C["36"]
    C -->|×3| D["108"]
    D -->|×3| E["324"]

The nth Term Formula (Geometric)

Formula: aₙ = a₁ × r⁽ⁿ⁻¹⁾

Where:

• aₙ = the term you want
• a₁ = first term
• r = common ratio
• n = which term

Example: Find the 6th term of 3, 6, 12, 24…

• a₁ = 3
• r = 2
• n = 6

Solution:

a₆ = 3 × 2⁽⁶⁻¹⁾
a₆ = 3 × 2⁵
a₆ = 3 × 32
a₆ = 96 ✓

Recursive Formulas: One Step at a Time

What’s a Recursive Formula?

A recursive formula tells you how to get the next term from the previous one. Like following breadcrumbs!

Arithmetic Recursive:

aₙ = aₙ₋₁ + d

“Next = Previous + common difference”

Geometric Recursive:

aₙ = aₙ₋₁ × r

“Next = Previous × common ratio”

Examples

Arithmetic (d = 5, a₁ = 2):

a₁ = 2
a₂ = a₁ + 5 = 2 + 5 = 7
a₃ = a₂ + 5 = 7 + 5 = 12
a₄ = a₃ + 5 = 12 + 5 = 17

Geometric (r = 2, a₁ = 3):

a₁ = 3
a₂ = a₁ × 2 = 3 × 2 = 6
a₃ = a₂ × 2 = 6 × 2 = 12
a₄ = a₃ × 2 = 12 × 2 = 24

Mean Insertion: Filling the Gaps

What is Mean Insertion?

Sometimes you know the first and last number, but need to find the numbers in between that keep the pattern!

Arithmetic Mean Insertion

Problem: Insert 3 arithmetic means between 2 and 18.

This means: 2, __, __, __, 18 (5 terms total)

Step 1: Find the common difference

d = (last - first) ÷ (number of terms - 1)
d = (18 - 2) ÷ (5 - 1)
d = 16 ÷ 4
d = 4

Step 2: Add d repeatedly

2, 6, 10, 14, 18 ✓

Single Arithmetic Mean

The arithmetic mean of two numbers is just their average!

Example: Find arithmetic mean between 10 and 20

Mean = (10 + 20) ÷ 2 = 15
Sequence: 10, 15, 20 ✓

Geometric Mean Insertion

Problem: Insert 2 geometric means between 2 and 54.

This means: 2, __, __, 54 (4 terms total)

Step 1: Find the common ratio

r = ⁿ√(last ÷ first)  where n = number of terms - 1
r = ³√(54 ÷ 2)
r = ³√27
r = 3

Step 2: Multiply by r repeatedly

2, 6, 18, 54 ✓

Single Geometric Mean

The geometric mean of two numbers a and b is √(a × b)

Example: Find geometric mean between 4 and 16

Mean = √(4 × 16) = √64 = 8
Sequence: 4, 8, 16 ✓

Quick Comparison Chart

Real Life Sequences

Arithmetic:

• Saving $50 every month (50, 100, 150, 200…)
• Climbing stairs (each step same height)
• Seats in a theater row

Geometric:

• Bacteria doubling (1, 2, 4, 8, 16…)
• Compound interest growth
• Folding paper (layers double!)

You’ve Got This! 🚀

Sequences are everywhere once you start looking. The key is simple:

• Same addition? → Arithmetic (find d)
• Same multiplication? → Geometric (find r)
• Need any term? → Use the formula
• Step by step? → Use recursive
• Fill gaps? → Mean insertion

Now go find patterns in the world around you!