# What is #sqrt(7 + sqrt(7 - sqrt (7 + sqrt (7 - sqrt(7 + ...... ∞#?

##### 3 Answers

#### Explanation:

Let

where we constrain our solution to be positive since we are taking only the positive square root i.e.

Where this time we constrain the left hand side to be positive, since we only want the positive square root i.e.

where we have eliminated the possibility the

Again squaring both sides we have

The expression in the repeated square roots is the original expression for

or

Trial solutions of this equation are

Using the quadratic formula on the third factor

The four roots of the polynomial are therefore

Another way

#### Explanation:

I like to discuss a tricky way to have a solution at a glance on the problem of repeated square roots like the following

where

TRICK

If 1 is subtracted from the given Number

when r =

when r = 3 the factor of (3-1)= 2 = 1.2 and 2 is the answer

when r = 7 the factor of (7-1) =6= 2.3 and 3 is the answer

and so on.......

**Explanation**

Taking

Squaring both sides

Again Squaring both sides

putting r =

if we put x = m in the LHS of this equation the LHS becomes

LHS =

the equation is satisfied.

**Hence m is the answer**

let's put

We can easily see that

So let's solve the equation:

This is not a trivial equation to be solved. One of the other persons that answered the question referred the solution 3. If you try it, you will see it's true.