Question

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

Answer 1

`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(n+1)=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(x-1) = 12`

Now let `x = n + 1`
Then `n(n+1) = 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:

`1rArrsqrt(72+sqrt(72+sqrt(72+sqrt(72+sqrt(72....)))))`

`Solutionrarr9`

`2rArrsqrt(30+sqrt(30+sqrt(30+sqrt(30+sqrt(30....)))))`

`Solutionrarr6`

And wait!!!

If you see a question like `sqrt(72-sqrt(72-sqrt(72-sqrt(72-sqrt(72....)))))`

`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!

Answer 2

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

`rarrsqrt(12+x)=x`

Square both sides

`rarr12+x=x^2`

`rarrx^2-x-12=0`

When we simplify this, we get

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

From this, we get, `x=4 and -3`. We need a positive value, so it is 4.