I'll go over the Fibonacci sequence, since that's what you asked about.

\( F_n = n^{th} \) term

The first term would be called \( F_1 \), the second term would be called \( F_2\), the third term would be called \( F_3 \), and so on.

The formula for getting any term of the Fibonacci sequence is this:

\( F_n = F_{n-1} + F_{n-2}\)

There is an \( n \) in the equation to express the exciting fact that this is a pattern that is true for any Fibonacci number!

So \( F_{n-1} \) means the term that came before this one (for example, \( F_1 \), which equals \( F_{2-1} \), comes before \( F_2 \)) . In the same way, \( F_{n-2} \) refers to the term before the previous term (for example, \( F_1 \), which equals \( F_{3-2} \), is the term before the term before \( F_3 \)).

The formula tells us this: that to get any term, you add the previous term to the previous-previous term.

The first two terms of the Fibonacci sequence are defined to be 1.

\( F_1 = 1 \)

\( F_2 =1 \)

Using the formula, we find that the third term is the first term plus the second term:

\( F_3 = F_1 + F_2 \)

\( F_3 = 1+1 \)

\( F_3 = 2 \)

The formula tells us that the fourth term is equal to the second term plus the third term:

\( F_4 = F_2 + F_3 \)

\( F_4 = 1+2 \)

\( F_4 = 3 \)

And the fifth term is equal to the third term plus the fourth term:

\( F_5 = F_3 + F_4 \)

\( F_5 = 2+3 \)

\( F_5 = 5 \)

If you write all the terms in a line, it's easy to write each successive term by simply adding the previous two:

\( \textcolor{red}{1 \text{ } 1} \text{ } 2 \)

\( 1 \text{ } \textcolor{red}{1 \text{ } 2} \text{ } 3 \)

\( 1 \text{ } 1 \text{ } \textcolor{red}{2 \text{ } 3} \text{ } 5 \)

\( 1 \text{ } 1 \text{ } 2 \text{ } \textcolor{red}{3 \text{ } 5} \text{ } 8 \)

\( 1 \text{ } 1 \text{ } 2 \text{ } 3 \text{ } \textcolor{red}{5 \text{ } 8} \text{ } 13 \)

\( 1 \text{ } 1 \text{ } 2 \text{ } 3 \text{ } 5 \text{ } \textcolor{red}{8 \text{ } 13} \text{ } 21 \)

The Fibonacci numbers are even more cool because they appear in nature, in the number of petals of flowers that grow in radial layers of certain plants, like this:

It also comes up in the spirals of snails, the breeding of rabbits, and more. Even more neat is the fact that the ratio of two numbers that are side-by-side in the Fibonacci sequence is equal to the golden ratio!

I hope this helps! We're happy to answer any questions you might have, and we'll do our best to make this course the best possible way to learn conceptual math!

Happy Learning,

The Daily Challenge Team

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