Fibonacci Sequence — Pattern, Formula & Golden Ratio

In the Fibonacci sequence each term is the sum of the two before it. Start with two values, keep adding the previous two, and the whole sequence unfolds: $0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\dots$

The formula and what the symbols mean

Using the common convention $F_0 = 0$ and $F_1 = 1$, the rule is the recurrence

Here $F_n$ is the $n$-th term, and each new term is built from the two earlier terms $F_{n-1}$ and $F_{n-2}$, not from a single direct rule applied once. The 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 check the indexing before comparing answers.

Why consecutive ratios approach the golden ratio

For positive Fibonacci terms, the ratio of consecutive terms moves toward the golden ratio:

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 $n$ grows.

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

where

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

Worked example: find F_8 and check a ratio

Start with

then build forward one step at a time:

So

Now compare a consecutive-term ratio:

which is already close to $\phi \approx 1.618$. The Fibonacci numbers are integers, yet ratios of consecutive terms drift toward the golden ratio. (As a smaller-$n$ contrast, $\frac{5}{3} \approx 1.667$ is further off, exactly as expected.)

Practice it yourself, then check

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

Check: continuing the build gives $F_9 = 34$ and $F_{10} = 55$, so $\frac{55}{34} \approx 1.618$. The ratio settles toward $\phi$ remarkably fast.

Calculation traps

Mixing up the starting index. If one source starts with $F_0 = 0, F_1 = 1$ and another with $F_1 = 1, F_2 = 1$, the same label can name different numbers. Check the convention first.

Thinking the ratio is always exactly $\phi$. Early ratios such as $\frac{5}{3} \approx 1.667$ are only approximations; the ratio approaches $\phi$ for large $n$.

Using the recurrence without two starting values. The rule needs two initial terms, or 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.

Where the Fibonacci sequence is used

It appears in counting problems where each case is built from the two earlier cases, and it is 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 shows up so often in math courses.