What is sqrt(12+sqrt(12+sqrt(12+sqrt(12+sqrt(12....)))))?

Feb 21, 2016

$4$

Explanation:

There is a really interesting math trick behind it.

If you see a question like this take out the number inside it (in this case is $12$)

Take consecutive numbers such as:

$n \left(n + 1\right) = 12$

Always remember that the answer is $n + 1$

This is true because if you let the infinite nested radical function = x then realise that x is also also under the first root sign as:

$x = \sqrt{12 + x}$

Then, squaring both sides: ${x}^{2} = 12 + x$
Or: ${x}^{2} - x = 12$
$x \left(x - 1\right) = 12$

Now let $x = n + 1$
Then $n \left(n + 1\right) = 12$ With the answer to the infinite nested radical function (x) being equal to $n + 1$

If you solve it you get $n = 3$ and $n + 1 = 4$

So,

The answer is $4$

Practice problems:

$1 \Rightarrow \sqrt{72 + \sqrt{72 + \sqrt{72 + \sqrt{72 + \sqrt{72. \ldots}}}}}$

$S o l u t i o n \rightarrow 9$

$2 \Rightarrow \sqrt{30 + \sqrt{30 + \sqrt{30 + \sqrt{30 + \sqrt{30. \ldots}}}}}$

$S o l u t i o n \rightarrow 6$

And wait!!!

If you see a question like $\sqrt{72 - \sqrt{72 - \sqrt{72 - \sqrt{72 - \sqrt{72. \ldots}}}}}$

$n$ is the solution (in this case is $8$)

Problems to solve on your own

sqrt(1056+sqrt(1056+sqrt(1056+sqrt(1056+sqrt(1056....))))

sqrt(110+sqrt(110+sqrt(110+sqrt(110+sqrt(110....))))

Better luck!

Oct 28, 2017

There is an other method to solve this

Explanation:

First of all, consider the whole equation equals $x$

color(brown)(sqrt(12+sqrt(12+sqrt(12....)))=x

We can also write it as

color(brown)(sqrt(12+x)=x

As, the $x$ is nested into it. Solve it

$\rightarrow \sqrt{12 + x} = x$

Square both sides

$\rightarrow 12 + x = {x}^{2}$

$\rightarrow {x}^{2} - x - 12 = 0$

When we simplify this, we get

color(green)(rArr(x+3)(x-4)=0

From this, we get, $x = 4 \mathmr{and} - 3$. We need a positive value, so it is 4.