Nested Roots

Algebra Level 1

F i n d t h e v a l u e o f 6 + 6 + 6 + 6 + . . . . . . . . Find\quad the\quad value\quad of\quad \\ \sqrt { 6+\sqrt { 6+\sqrt { 6+\sqrt { 6+........ } } } }


The answer is 3.

Rajdeep Dhingra by Rajdeep Dhingra , 20, India

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1 solution

Rajdeep Dhingra
Oct 16, 2014

L e t 6 + 6 + 6 + . . . . . . b e = x s o 6 + x = x S q u a r i n g b o t h s i d e s 6 + x = x 2 x 2 − x − 6 = 0 s o l v i n g x = 1 ± 1 2 − 4 ( − 6 ) 2 x = 3 o r − 2 R o o t s c a n ′ t b e n e g a t i v e H e n c e x = 3 Let\quad \sqrt { 6+\sqrt { 6+\sqrt { 6+...... } } } be\quad =\quad x\\ so\\ \sqrt { 6\quad +\quad x } \quad =\quad x\\ Squaring\quad both\quad sides\\ 6\quad +\quad x\quad =\quad { x }^{ 2 }\\ { x }^{ 2 }\quad -\quad x\quad -\quad 6\quad =\quad 0\\ solving\\ x\quad =\quad \frac { 1\quad \pm \quad \sqrt { { 1 }^{ 2 }-4(-6) } }{ 2 } \\ x\quad =\quad 3\quad or\quad -2\\ Roots\quad can't\quad be\quad negative\quad \\ Hence\quad x\quad =\quad 3\\ root root

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how x can be equal to root over 6+x??????

Ariful Ashiq - 5 years, 9 months ago

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Since ,X=√6+√6..... We see that in,√6+X=X ------1 Pit the value of X in equation 1. ✔️6+√6...=✔️6+√6

SayHi2God Rohit - 1 year, 1 month ago

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