Coincidence?

Algebra Level 3

What is $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1...}}}}$ equal to? (4 significant figures)

1.876 1.618 1 1.732 1.414 3.142

3 solutions

Adam Zaim
Sep 10, 2015

Let $\sqrt { 1+\sqrt { 1+\sqrt { 1+\dots } } } =S$

Then substitute to itself...

$\sqrt { 1+S } =S$

$1+S={ S }^{ 2 }$

$0=S^2-S-1$

$S=\frac { -b\pm \sqrt { b^2 -4ac} }{ 2a }$

$S=\frac { 1\pm \sqrt { 1-4(1)(-1) } }{ 2 }$

$S=\frac { 1\pm \sqrt { 5 } }{ 2 }$

$S=-0.6180$ or $S=1.618$

But $\sqrt { 1-0.6180 } \neq -0.6180$ because the square root of any positive number cannot equal to a negative number. So...

$S=1.618$

Same Method :D

Syed Baqir - 5 years, 9 months ago

I want someone to put a couple of options that has negative numbers. Just to make it a little bit confused. :3

Adam Zaim - 5 years, 9 months ago

That would be great idea !

Syed Baqir - 5 years, 9 months ago

Not a bad idea. But the answer to a square root of a positive number can't be negative, so i doubt anyone would pick the negative options.

Gaurav Agarwal - 5 years, 9 months ago

Yellow Tomato
Sep 9, 2015

The golden ratio

$\frac{1 + \sqrt5}{2}$

John Gilling
Sep 9, 2015

Let the quantity equal x. Then, $x^2-1=x$ , and the quadratic formula gives $x=\frac{1+\sqrt{5}}{2}$ .