This is a nested radical.
a = √(1+√(1+√(1+......
informally
a = √(1+a)
a² = 1+a
a²-a-1=0
quadratic formula
a=(1±√5)/2
(1+√5)/2 ≈ 1.618 = φ
(1-√5)/2 ≈ -.618 = 1-φ
assuming the continued fractions converges
then a = sqrt(1+a)
a^2 = a+1
a^2 - a - 1 = 0
a = (1+-sqrt(1-4*-1))/2 = ( 1+sqrt(5))/2 = 1.618
you mean sqrt(1+sqrt(1+sqrt(1+...
lets see if that equals x then
x^2 = 1+ x
solve the quadratic and you get phi
(but be aware this is formal reasoning, not rigorous)
√1 = 1
so every time u put √1 like u put 1
so √1 + √1 = 1 +1 = 2 ... etc.
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Verified answer
This is a nested radical.
a = √(1+√(1+√(1+......
informally
a = √(1+a)
a² = 1+a
a²-a-1=0
quadratic formula
a=(1±√5)/2
(1+√5)/2 ≈ 1.618 = φ
(1-√5)/2 ≈ -.618 = 1-φ
assuming the continued fractions converges
then a = sqrt(1+a)
a^2 = a+1
a^2 - a - 1 = 0
a = (1+-sqrt(1-4*-1))/2 = ( 1+sqrt(5))/2 = 1.618
you mean sqrt(1+sqrt(1+sqrt(1+...
lets see if that equals x then
x^2 = 1+ x
solve the quadratic and you get phi
(but be aware this is formal reasoning, not rigorous)
√1 = 1
so every time u put √1 like u put 1
so √1 + √1 = 1 +1 = 2 ... etc.