Question

What is the value of $\sqrt {6 + \sqrt {6 + \sqrt {6+ ...}}}$?

Answer 1

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

Answer is 3 take the square roots of all and add them