Week 1: Patterns, Sequences, and Thinking Like a Mathematician

Warm-Up: What Comes Next?

Look at each sequence below. Can you figure out the pattern and predict the next three numbers?

2, 4, 6, 8, ___, ___, ___
1, 4, 9, 16, 25, ___, ___, ___
1, 1, 2, 3, 5, 8, ___, ___, ___

Core Lesson: The Language of Sequences

What is a Sequence?

A sequence is an ordered list of numbers that follows a rule. Mathematicians love sequences because they're everywhere — in nature, music, architecture, and computer science.

💡 Key Concept

Every sequence has a rule (or formula) that tells you how to find any term. If you know the rule, you can find the 10th term, the 100th term, or even the millionth term — without writing them all out.

Arithmetic Sequences

An arithmetic sequence adds the same number each time. That number is called the common difference (d).

| Term | Value | How we got it | |--------|-------|-------------------| | 1st | 3 | start | | 2nd | 7 | 3 + 4 | | 3rd | 11 | 7 + 4 | | 4th | 15 | 11 + 4 | | nth | ? | 3 + 4 × (n - 1) |

The general formula for the nth term is:

a(n) = a(1) + d × (n - 1)

So for our sequence: a(n) = 3 + 4(n - 1) = 4n - 1

What is the 50th term of the sequence 3, 7, 11, 15, ...?

Geometric Sequences

A geometric sequence multiplies by the same number each time. That number is called the common ratio (r).

| Term | Value | How we got it | |--------|-------|-------------------| | 1st | 2 | start | | 2nd | 6 | 2 × 3 | | 3rd | 18 | 6 × 3 | | 4th | 54 | 18 × 3 | | nth | ? | 2 × 3^(n-1) |

💡 Key Concept

Arithmetic = adding the same thing each time (grows linearly)

Geometric = multiplying by the same thing each time (grows exponentially — much faster!)

Which of these is a geometric sequence?

The Fibonacci Sequence

The Fibonacci sequence is special: each term is the sum of the two before it.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

This sequence shows up everywhere in nature:

The number of petals on flowers (often 3, 5, 8, 13, or 21)
The spiral pattern of sunflower seeds
The branching of trees
The shape of seashells and galaxies

The Fibonacci Sequence in Nature — TED-Ed

Hands-On Lab: Your First Python Program

It's time to write code! We'll use Python to generate sequences automatically.

Getting Started

Open your browser and go to replit.com. Create a free account if you don't have one, then create a new Python project.

Program 1: Generate an Arithmetic Sequence

Type this into the editor and hit Run:

# Arithmetic sequence generator
start = 3
difference = 4

for i in range(1, 21):
    term = start + difference * (i - 1)
    print(f"Term {i}: {term}")

Try changing start and difference to create your own sequences. What happens if the difference is negative?

Program 2: The Fibonacci Sequence

Now let's generate Fibonacci numbers:

# Generate the first 50 Fibonacci numbers
a, b = 1, 1

for i in range(50):
    print(f"Fibonacci #{i + 1}: {a}")
    a, b = b, a + b

Notice how fast the numbers grow! By the 50th term, the number is over 12 billion.

Program 3: Make It Interactive

Let's let the user choose what kind of sequence to generate:

# Interactive sequence generator
print("=== Sequence Generator ===")
print("1. Arithmetic sequence")
print("2. Geometric sequence")
print("3. Fibonacci sequence")

choice = input("\nPick one (1, 2, or 3): ")

if choice == "1":
    start = int(input("Starting number: "))
    diff = int(input("Common difference: "))
    n = int(input("How many terms? "))
    for i in range(n):
        print(start + diff * i)

elif choice == "2":
    start = int(input("Starting number: "))
    ratio = int(input("Common ratio: "))
    n = int(input("How many terms? "))
    value = start
    for i in range(n):
        print(value)
        value = value * ratio

elif choice == "3":
    n = int(input("How many Fibonacci numbers? "))
    a, b = 1, 1
    for i in range(n):
        print(a)
        a, b = b, a + b

else:
    print("Invalid choice!")

✨ Tip

Python basics you just learned:

for i in range(n) — repeat something n times
print() — display output
input() — ask the user for input
if/elif/else — make decisions
int() — convert text to a number
f"Term {i}: {term}" — format strings with variables

Challenge: Gauss's Trick

The story goes that when the mathematician Carl Friedrich Gauss was 10 years old, his teacher asked the class to add up all the numbers from 1 to 100. While his classmates started adding one by one, Gauss found the answer in seconds.

Your challenge: Can you figure out Gauss's trick? Find a formula for:

1 + 2 + 3 + 4 + ... + n = ?

Hint: What do you get if you add the first number and the last number? The second number and the second-to-last number?

Bonus: Write a Python program that checks Gauss's formula against a brute-force loop:

n = 100

# Method 1: Brute force (add them one by one)
total = 0
for i in range(1, n + 1):
    total = total + i
print(f"Brute force: {total}")

# Method 2: Gauss's formula
formula = n * (n + 1) // 2
print(f"Gauss formula: {formula}")

# Do they match?
print(f"Same answer? {total == formula}")

Resources

Python Basics — W3Schools — reference for Python syntax
Replit — online code editor
OEIS (Online Encyclopedia of Integer Sequences) — look up any sequence and find its rule
Fibonacci in Nature — Math is Fun — more on Fibonacci

Patterns, Sequences, and Thinking Like a Mathematician

Materials for this lesson