Question

root over 2+root over 2 +root over 2 upto infinity

Answer 1

$\infty \sqrt{2}$

Answer 2

Answer:

2 or 1

Step-by-step explanation:

Given,

$\sqrt[]{2+\sqrt[]{2+\sqrt[]{2+\sqrt[]{2+\sqrt[]{2+}\sqrt[]{2+} ........infinite } } } }$

So,

Let x = $\sqrt[]{2+\sqrt[]{2+\sqrt[]{2+\sqrt[]{2+\sqrt[]{2+}\sqrt[]{2+} ........infinite } } } }$

⇒ $x$ = $\sqrt[]{2+x}$

squaring on both sides

⇒ $x^{2} = 2+x$

Transpose $2+x$ form  R.H.S to L.H.S

⇒$x^{2} -x-2$ = 0

use formula

$\frac{-b±\sqrt[]{b^{2}-4ac } }{2a}$ where A bar is 1

a = 1, b= -1, c =- 2

$\frac{-(-1)±\sqrt[]{(-1)^{2}-4(1)(-2) }}{2(1)}$

= 1±3/2

x = 1+3/2

= 4/2 = 2

(or)

x = 1-3/2

= 2/2

= 1

∴ x = 2 or 1