Question

The value of$\sqrt{6+\sqrt{6+\sqrt{6}}}$.... is
(a)4
(b)3
(c)−2
(d)3.5

Answer 1

SOLUTION :  

Option (b) is correct :  3

Let x = [√6 + √6 + √6 + √6 + …..]

x = √(6 + x)...........(1)

On squaring both sides,

x² = 6 + x  

x² - x - 6 = 0

x² - 3x + 2x - 6 = 0

[By middle term splitting]

x(x - 3) + 2(x - 3) = 0

(x - 3) (x + 2 ) = 0

(x - 3) = 0 (x + 2 ) = 0

x = 3 or x = - 2  

On putting x = 3 in eq 1,

x = √(6 + x)

3 = √(6 +3)

3 = √9 = 3  

This is possible  

On putting x = - 2 in eq 1,

x = √(6 + x)

-2 = √(6 - 2)

- 2 = √4 = 2

This is not possible .

Hence , the value is 3 .

HOPE THIS ANSWER WILL HELP YOU...

Answer 2

Solution :

Let $\sqrt{6+\sqrt{6+\sqrt{6}}} = a$

=> $\sqrt {6+a}=a$

=> $(\sqrt{6+a})^{2})=a^{2}$

=> 6 + a = a²

=> a² - a - 6 = 0

Splitting the middle term , we

get

=> a² - 3a + 2a -6 = 0

=> a( a - 3 ) + 2( a - 3 ) = 0

=> ( a - 3 )( a + 2 ) = 0

=> a - 3 = 0 or a + 2 = 0

=> a = 3 or a = -2

Therefore ,

Value of $\sqrt{6+\sqrt{6+\sqrt{6}}}$... is 3

Option ( b ) is correct.

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