Fibonacci Sequence — Pattern, Formula & Golden Ratio

The Fibonacci sequence is a number pattern in which each term is the sum of the two before it. Using the common convention $F_0 = 0$ and $F_1 = 1$, the rule is

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

so the sequence begins

$$0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\dots$$

If you only need the main idea, it is this: start with two values, then keep adding the previous two to get the next one.

What the Fibonacci sequence is

The Fibonacci sequence is defined by a recurrence relation. That means each new term is built from earlier terms, not from a single direct rule you apply once.

This sequence depends on the starting convention. Many textbooks use $F_0 = 0$ and $F_1 = 1$. Others use $F_1 = 1$ and $F_2 = 1$. The number pattern is the same, but the labels shift, so always check the indexing before comparing answers.

Fibonacci sequence formula

The main formula is the recurrence:

$$F_n = F_{n-1} + F_{n-2}$$

It says each term comes from the previous two. For example,

$$F_5 = F_4 + F_3 = 3 + 2 = 5$$

There is also a closed form, often called Binet's formula. Under the convention $F_0 = 0$ and $F_1 = 1$,

$$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$$

where

$$\phi = \frac{1+\sqrt{5}}{2}, \qquad \psi = \frac{1-\sqrt{5}}{2}$$

For most students, the recurrence is the better place to start. Binet's formula is useful because it connects Fibonacci numbers to powers and to the golden ratio, but you do not need it to generate terms.

Why Fibonacci ratios approach the golden ratio

For positive Fibonacci terms, the ratio of consecutive terms gets closer to the golden ratio:

$$\phi = \frac{1+\sqrt{5}}{2} \approx 1.618$$

More precisely, if you look at

$$\frac{F_{n+1}}{F_n}$$

for larger and larger $n$ with $F_n \ne 0$, the ratio approaches $\phi$. That does not mean every ratio equals $\phi$. It means the ratios converge to $\phi$ as $n$ gets larger.

Worked example: find $F_8$

Use the recurrence to find $F_8$ and then check a nearby ratio.

Start with

$$F_0 = 0,\qquad F_1 = 1$$

Then build forward one step at a time:

$$F_2 = 1,\quad F_3 = 2,\quad F_4 = 3,\quad F_5 = 5,\quad F_6 = 8,\quad F_7 = 13,\quad F_8 = 21$$

So

$$F_8 = 21$$

Now compare a consecutive-term ratio:

$$\frac{F_8}{F_7} = \frac{21}{13} \approx 1.615$$

This is close to

$$\phi \approx 1.618$$

That is the key connection: the Fibonacci numbers are integers, but ratios of consecutive terms move toward the golden ratio.

Common mistakes with the Fibonacci sequence

Mixing up the starting index

If one source starts with $F_0 = 0, F_1 = 1$ and another starts with $F_1 = 1, F_2 = 1$, the same term label can refer to different numbers. Always check the convention first.

Thinking the ratio is always exactly the golden ratio

The ratio $\frac{F_{n+1}}{F_n}$ approaches $\phi$ for large $n$, but early ratios are only approximations. For example, $\frac{5}{3} \approx 1.667$, which is not equal to $\phi$.

Using the recurrence without two starting values

The rule needs two initial terms. Without them, the sequence is not fully determined.

Treating every "growing pattern" as Fibonacci

A pattern is Fibonacci only if each term really is the sum of the previous two, under a stated starting convention. Similar-looking lists are not enough.

When the Fibonacci sequence is used

The Fibonacci sequence appears in counting problems where each case can be built from the two earlier cases. It is also a standard example in algebra, discrete math, algorithms, and proofs by induction.

It matters beyond this one topic because it teaches three ideas at once: recursive definition, closed form, and limiting behavior. That combination is why it appears so often in math courses.

Try your own version

Write the sequence through $F_{10}$, then compute $\frac{F_{10}}{F_9}$. Compare your result with $\phi \approx 1.618$.

If you want one more case after that, try your own version with a different index target and see how quickly the ratio settles down.