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

2 Answers
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(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!

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

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.