2 and its square root

Algebra Level 3

2 + 2 + 2 + 2 + 2 + 2 + … \sqrt { 2+\sqrt { 2 } +\sqrt { 2+\sqrt { 2 } +\sqrt { 2+\sqrt { 2 } +\ldots } } } The value of above expression is in the form a + b \sqrt { a } +b , where a a and b b are integers, find a + b a+b .


The answer is 3.

View wiki
Fredirick Estrella by Fredirick Estrella , 25, Philippines

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Rishabh Jain
Feb 25, 2016

x = 2 + 2 + 2 + 2 + 2 + 2 + . . . \color{#D61F06}{x}=\sqrt { 2+\sqrt 2+\color{#D61F06}{\sqrt { 2+\sqrt { 2 } +\sqrt { 2+\sqrt { 2 } +... } } }} = 2 + 2 + x = \sqrt{2+\sqrt 2+\color{#D61F06}{x}} ⟹ x 2 − x − ( 2 + 2 ) = 0 \implies x^2-x-(2+\sqrt 2)=0 ⟹ x = 1 + 9 + 4 2 2 \implies x=\dfrac{1+\sqrt{9+4\sqrt 2}}{2} = 1 + ( 2 2 + 1 ) 2 = 2 + 1 = \dfrac{1+(2\sqrt 2+1)}{2}=\sqrt 2+1 ∴ 2 + 1 = 3 \huge \therefore 2+1=\boxed{\color{#D61F06}{3}}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice and Colourful !!

Akshat Sharda - 5 years, 3 months ago

Log in to reply

T H A N K S ! ! \huge\mathbb{\color{#302B94}{THANKS!!}}

Rishabh Jain - 5 years, 3 months ago

Log in to reply

How do you write in such colourful font?

Ahmed K - 5 years, 3 months ago

Log in to reply

@Ahmed K – Do toggle latex.

Akshat Sharda - 5 years, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...