Question

$\sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + .......} } } } }$
plz solve with solution for brainlist​

Answer 1

Question

$\tt{ \sqrt{6 + \sqrt{6 + \sqrt{6 + \cdots} } }}$

Solution

$\tt{Let \: y = \sqrt{6 + \sqrt{6 + \sqrt{6 + \cdots \infty} } } - - - (1)}$

Implies,

$\tt{y = \sqrt{6 + ( \sqrt{6 + \sqrt{6 + \cdots} } } )}$

From equation (1),we write,

$\tt{ \longrightarrow \:y = \sqrt{6 + y}}$

Squaring on both sides :

$\longrightarrow \: \tt{ {y}^{2} = 6 + y } \\ \\ \tt{ \longrightarrow \: {y}^{2} - y - 6 = 0 } \\ \\ \tt{ \longrightarrow \: y {}^{2} - 3y + 2y - 6 = 0 } \\ \\ \tt{ \longrightarrow \: (y + 2)(y - 3) = 0 } \\ \\ \longrightarrow \: \boxed{ \boxed{\tt{y = - 2 \: or \: 3}}}$

Now,

y can't be a negative number as it is a square root and real number. Thus,y = 3

Therefore,

$\tt{ \sqrt{6 + \sqrt{6 + \sqrt{6 + \cdots} } } = 3}$

Answer 2

$\bold{\huge{\underline{\underline{\sf{AnsWer:}}}}}$

$\sf{\pink{Value\: of\:the\: infinite\: series\: is\:3}}$

$\bold{\large{\underline{\underline{\sf{StEp\:by\:stEp\:explanation:}}}}}$

$\sqrt{6 + \sqrt{6 +\sqrt{6 + \sqrt{6 +\sqrt{6 + .......}}}}}$

★ We notice that √6 occurs again and again. We infer that it is an infinite series.

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

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

★ From equation (i)

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

★ Squaring both sides,

→ $\tt{x^2\:=\:6+x}$

→ $\tt{x^2\:-\:6\:-\:x\:=\:0}$

★ We formed a quadratic equation in variable x.

→ $\tt{x^2\:-\:3x\:+\:2x\:-\:6\:=\:0}$

→ $\tt{x(x-3) \:+\:2\:(x-3)\:=\:0}$

→ $\tt{(x-3)\:\:or\:\:(x+2)\:\:=\:0}$

→ $\tt{x-3=0\:\:or\:\:x+2=0}$

→ $\tt{x=3\:\:or\:\:x\:=\:-2}$

★ Since x is a square root and a real number, therefore value of x must be positive.

$\tt{\therefore{x\:=\:-2\:is\:not\:acceptable}}$

$\bold{\boxed{\red{\tt{\therefore{x\:=\:3\:is\:\:the\:value\:of\:the\:given\:infinte\:series}}}}}$

★ Shortcut trick :

[Trick applicable only for numbers whose factor difference is 1]

To solve in a short way :

★ Find the factors of the number given in the question whose difference is 1.

In the above question, we have √6.

Ignore the square roots, factors of 6 are : 3, 2

Difference between the factors :3 - 2 = 1

★ Since the question contains + sign, we have to go for the larger factor of the given number, ignoring the smaller one.

So the factor of 6, i.e 3 & 2, the larger of the two numbers is 3.

And we are done with our solution!

Either way we got the same value,3.

★ If the question contains - (minus) sign, choose the smaller factor ignoring the larger one.

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