The value of \(\sqrt{2+\sqrt{2+\sqrt{2 \,+............}}}\) is......

1.	The value of \(\sqrt{2+\sqrt{2+\sqrt{2 \,+............}}}\) is...........A) 3 B) 4C) 2 D) 8
Answer» Correct option is (C) 2 Let x = \(\sqrt{2+\sqrt{2+\sqrt{2\,+......}}}\) \(\Rightarrow x=\sqrt{2+x}\) \(\Rightarrow x^2=2+x\) (By squaring both sides) \(\Rightarrow x^2-x-2=0\) \(\Rightarrow x^2-2x+x-2=0\) \(\Rightarrow\) x (x - 2) + 1 (x - 2) = 0 \(\Rightarrow\) (x + 1) (x - 2) = 0 \(\Rightarrow\) x = -1 or x = 2 \(\because\) \(\sqrt{2+\sqrt{2+\sqrt{2\,+......}}}\) \(>\sqrt2>0\) \(\therefore\) \(x\neq-1\) \(\therefore\) x = 2 Hence, \(\sqrt{2+\sqrt{2+\sqrt{2\,+......}}}=2\) Correct option is C) 2

The value of \(\sqrt{2+\sqrt{2+\sqrt{2 \,+............}}}\) is...........A) 3 B) 4C) 2 D) 8

Answer»

Correct option is (C) 2

Let x = \(\sqrt{2+\sqrt{2+\sqrt{2\,+......}}}\)

\(\Rightarrow x=\sqrt{2+x}\)

\(\Rightarrow x^2=2+x\) (By squaring both sides)

\(\Rightarrow x^2-x-2=0\)

\(\Rightarrow x^2-2x+x-2=0\)

\(\Rightarrow\) x (x - 2) + 1 (x - 2) = 0

\(\Rightarrow\) (x + 1) (x - 2) = 0

\(\Rightarrow\) x = -1 or x = 2

\(\because\) \(\sqrt{2+\sqrt{2+\sqrt{2\,+......}}}\) \(>\sqrt2>0\)

\(\therefore\) \(x\neq-1\)

\(\therefore\) x = 2

Hence, \(\sqrt{2+\sqrt{2+\sqrt{2\,+......}}}=2\)

Correct option is C) 2