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Fibonacci Numbers (Sequence)

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Fibonacci Numbers (Sequence)

Study Guide

Key Definition

The Fibonacci sequence is defined recursively: $F_0 = 0$, $F_1 = 1$, $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$.

Important Notes

Fibonacci numbers appear in nature, art, and architecture.
The ratio of successive Fibonacci numbers approximates the Golden Ratio: $\phi \approx 1.618$.
Fibonacci numbers are defined recursively.
The sequence starts with $0, 1$.
Each subsequent number is the sum of the two preceding ones.

Mathematical Notation

$+$ represents addition

$\phi$ is the Golden Ratio

$\sqrt{}$ represents the square root

$F_n$ denotes the nth Fibonacci number

Remember to use proper notation when solving problems

Why It Works

The Fibonacci sequence grows by adding the previous two numbers, which creates a unique pattern with properties related to the Golden Ratio.

Remember

The Fibonacci sequence begins with $0, 1$ and continues with the sum of the last two numbers.

Quick Reference

Fibonacci Formula:$F_n = F_{n-1} + F_{n-2}$

Understanding Fibonacci Numbers (Sequence)

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The Fibonacci sequence starts with $0, 1$, and each subsequent number is the sum of the two before it: $0 + 1 = 1$, $1 + 1 = 2$, etc.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the next number in the Fibonacci sequence: $1, 1, 2, 3, 5, 8$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are arranging flowers in a spiral pattern. The number of petals follows the Fibonacci sequence. If you start with $3$ petals in one layer followed by $5$ petals in the next, how many petals will be in the two subsequent layers?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a staircase with $10$ steps. Each step can be taken as a single step or as a pair (1 step or 2 steps at a time). How many distinct ways can you climb to the top?

Challenge Quiz

Single Choice Quiz

Advanced

If $F_{10} = 55$, what is $F_{12}$?

Recap

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