Problem#0

Level 2

x = 1 + 1 1 + 1 1 + 1 1 + 1 1 + ⋯ \large {x=1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\cdots } } } } } Find x correct to 5 decimal places


The answer is 1.61803.

Sumukh Bansal by Sumukh Bansal , 16, India

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1 solution

The Beginner
Nov 8, 2017

x = 1 + 1 1 + 1 1 + 1 1 + 1 1 + ⋯ \large {x=1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\cdots } } } } }

The above expression can also be written as x 2 − x − 1 = 0 x^2-x-1=0 . This gives us our answer as x = 5 + 1 2 x=\dfrac{\sqrt{5} +1}{2} . Here we have neglected the negative solution.

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