Question

$\sqrt{6 + \sqrt{6 + \sqrt{6 + ....} } }$
what is value of this question​

Answer 1

Given that, $\sqrt{6 + \sqrt{6 + \sqrt{6 + .... \: \infty } } }$

We have to find the value of the above equation.

Let x = $\sqrt{6 + \sqrt{6\: +\: \sqrt{6\: +\: .... \: \infty } } }$ ---- (eq 1)

$\implies\:x\:=\:\sqrt{6\: +\: \bigg( \sqrt{6 \:+\: \sqrt{6 \:+\: .... } } \bigg) }$

From (eq 1) we can write it as,

$\implies\:x\:=\:\sqrt{6 \:+ \:x }$

Now, squaring on both sides,

$\implies\:(x)^2\:=\:(\sqrt{6\: + \:x })^2$

$\implies\:x^2\:=\:6\:+\:x$

$\implies\:x^2\:-\:x\:-\:6\:=\:0$

The above equation is in the form ax² + bx + c = 0

Now, solve it by splitting the middle term.

$\implies\:x^2\:-\:3x\:+\:2x\:-\:6\:=\:0$

$\implies\:x(x\:-\:3)\:+2(x\:-\:3)\:=\:0$

$\implies\:(x\:+\:2)\:(x\:-\:3)\:=\:0$

$\implies\:x\:=\:-2,\:+3$

(-2 neglected)

So,

$\implies\:x\:=\:3$

Therefore, $\sqrt{6\: + \sqrt{6 \:+ \sqrt{6 \:+ .... \: } } } \:=\:3$

Answer 2

Question :---

$\sqrt{6 + \sqrt{6 + \sqrt{6 + - - - - \infty } } }$

Solution :---

Let Question is Equal to x..

$\sf \: x = \sqrt{6 + \sqrt{6 + \sqrt{6 + - - - - \infty } } } \\ \\ \rm \: squaring \: both \: sides \: we \: get \\ \\ \sf {x}^{2} = 6 + \sqrt{6 + \sqrt{6 + - - - - \infty } } \: \\ \\ \bf \: putting \: value \: of \: x \: now \\ \\ \sf \: {x}^{2} = 6 + x$

→ x² - x - 6 = 0

Splitting the Middle term now,

→ x² - 3x +2x - 6 = 0

→ x(x-3)+2(x-3) = 0

→ (x-3)(x+2) = 0

putting both Equal to zero we get,

→ x - 3 = 0. and x +2 = 0

→ x = 3 x = (-2)

Since, √6+√6+√6________ is positive , Hence, value of x is 3.. (Ans)..

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✪✪ Shortcut ✪✪ :----

Given value in Question is √6 + √6 + √6________ upto infinty..

So, we will break inside root value in 2 such part that, differnce between them is 1 or, we can say that, We will check two consecutive Multiples ...

Given = 6

→ 6 can be written as = 2 * 3

Now, if Question have + sign inside root , than our answer will be larger Multiple , and if Question have negative sign , our answer will be smaller value..

→ we have + sign in Question...

So, our Answer will be 3.

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If Question was √6 - √6 - √6________ upto infinty..

Than our Answer would be 2..

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Lets take some Examples :---

1) √20 + √20 + √20 ____________________ upto infinity .. ?

Ans:--- 20 = 4*5 , + sign means answer is 5..(Larger Multiple .)

2) √42 - √42 - √42 ___________________ upto infinity .?

Ans:--- 42 = 6*7, - sign means answer is 6.. (Smaller Mulitple).