Fibonacci Numbers and the Golden Ratio

01 · Begin with the pattern

The Fibonacci sequence begins

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Each new term is the sum of the two terms before it. Ratios of neighboring terms—(2/1, 3/2, 5/3, 8/5, 13/8)—move above and below a number close to 1.618.

02 · Build the golden ratio

Suppose the neighboring-term ratio has settled near (x). Dividing the Fibonacci rule by the current term gives:

x = 1 + 1/x

Multiply by (x):

x² = x + 1

The positive root is

φ = (1 + √5)/2 ≈ 1.618

03 · Understand Ella's equation

Let a=1/φ≈0.618. Starting with

1 = 1/a − a

multiply by a ≠ 0:

a² + a − 1 = 0

Therefore a=(−1±√5)/2. The positive solution is a=(√5−1)/2≈0.618; the other solution is −1.618….

The key identity is:

1/a = a + 1

At this special number, the transformations “take the reciprocal” and “add one” produce the same result. This does not merge addition with multiplication: the additive identity remains 0, and the multiplicative identity remains 1.

04 · See the self-similarity

1Start with φ.

2Remove one whole unit.

3The remainder is 1/φ.

4The whole-to-part ratio repeats.

A golden rectangle behaves the same way: removing its largest square leaves a smaller rectangle with the same shape. An infinite continued fraction repeats too:

φ = 1 + 1/(1 + 1/(1 + 1/(…)))

05 · From easy to hard

Warm-up

The next terms after (5,8,13) are (21) and (34).

Core algebra

Because (φ²=φ+1), higher powers collapse to linear expressions:

φ³ = 2φ + 1, φ⁴ = 3φ + 2, φ⁵ = 5φ + 3

The coefficients are Fibonacci numbers. In general, (φ^n=F_nφ+F_{n-1}).

AMC 10 extension

A self-similar infinite expression can be named with one variable. If (y=1+1/(1+1/(1+\cdots))), its repeating tail is also (y), so (y=1+1/y) and (y=φ).

06 · What the charting ratios mean

Ratio	Mathematical source	Practical role
0.618	1/φ	A standardized retracement reference
0.786	√0.618	A deeper reference zone
0.886	√0.786	An even deeper reference zone
0.500, 0.705	Trading conventions	Useful measuring choices, not classical Fibonacci ratios

These levels are coordinates, not causes. A beautiful ratio does not force price—or any real system—to reverse.

07 · Your turn

1 · Continue 2, 3, 5, 8, 13 for two terms.

21, 34. Add the previous two terms each time.

2 · If a>0 and 1=1/a−a, find a²+a.

1. Multiply the equation by a.

3 · Simplify φ⁵.

5φ+3. Repeatedly replace φ² with φ+1.

4 · Express φ⁸ as mφ+n.

21φ+13. The coefficients are F₈ and F₇.

5 · Solve z=2+1/z for z>0.

1+√2. Solve z²−2z−1=0. It is not φ because the fixed-point equation adds 2 rather than 1.