The Golden Ratio (Continued Fraction)
Evaluate an infinite nested fraction
x = 1 + 1 / (1 + 1 / (1 + ...)) = ϕ(1.618…)φ (1.618...)
The denominator repeats the whole expression, so the tail is x again.
Multiply through by x: x2=x+1x² = x + 1, which is x2−x−1=0x² − x − 1 = 0.
The positive root is (1+5)÷2(1 + √5) ÷ 2, about 1.618 — the golden ratio.
Your turn
One to try in your head — tap what you get.
Value of 1+(1+(1+…)1)11 + 1/(1 + 1/(1+...))
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