What is #sqrt(7 + sqrt(7 - sqrt (7 + sqrt (7 - sqrt(7 + ...... ∞#?
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
Trial solutions of this equation are
Using the quadratic formula on the third factor
The four roots of the polynomial are therefore
I like to discuss a tricky way to have a solution at a glance on the problem of repeated square roots like the following
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.......
Squaring both sides
Again Squaring both sides
putting r =
if we put x = m in the LHS of this equation the LHS becomes
the equation is satisfied.
Hence m is the answer
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.