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

3 Answers
Mar 9, 2016

3

Explanation:

Let

x=sqrt(7+sqrt(7-sqrt(7+sqrt(7-sqrt(7-sqrt(7+...oo

where we constrain our solution to be positive since we are taking only the positive square root i.e. x>=0. Squaring both sides we have

x^2=7+sqrt(7-sqrt(7+sqrt(7-sqrt(7-sqrt(7+...oo

=>x^2-7=sqrt(7-sqrt(7+sqrt(7-sqrt(7-sqrt(7+...oo

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

x^2-7>=0 => x>=sqrt(7) ~= 2.65

where we have eliminated the possibility the x<=-sqrt(7) using our first constraint.

Again squaring both sides we have

(x^2-7)^2=7-sqrt(7+sqrt(7-sqrt(7-sqrt(7+........oo

(x^2-7)^2-7=-sqrt(7+sqrt(7-sqrt(7-sqrt(7+........oo

The expression in the repeated square roots is the original expression for x, therefore

(x^2-7)^2-7=-x

or

(x^2-7)^2-7+x=0

Trial solutions of this equation are x=-2 and x=+3 which results in the following factorization

(x+2)(x-3)(x^2+x-7)=0

Using the quadratic formula on the third factor (x^2+x-7)=0 gives us two more roots:

(-1+-sqrt(29))/2 ~= 2.19 " and " -3.19

The four roots of the polynomial are therefore -3.19..., -2, 2.19..., and 3. Only one of these values satisfies our constraint x>=sqrt(7) ~= 2.65, therefore

x=3

Mar 10, 2016

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
sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+........oo

where r belongs to the following series
3,7,13,21,31............, the general term of which is given by
m^2-m+1 where m epsilon N and m>1

TRICK
If 1 is subtracted from the given Number m^2-m+1 the resulting number becomes m^2-m which is m(m-1) and which is nothing but the product of two consecutive number and larger one of these two will be the unique solution of the problem.

when r = m^2-m+1 the factor of m^2-m+1-1 = (m-1)m and m is the answer

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
x=sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+........oo
Squaring both sides
x^2= r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+........oo

x^2- r=sqrt(r-sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+........oo
Again Squaring both sides
(x^2- r)^2=r-sqrt(r+sqrt(r-sqrt(r+sqrt(r-sqrt(r+........oo
(x^2- r)^2-r=-x
(x^2- r)^2-r+x=0
putting r = m^2-m+1

(x^2- (m^2-m+1))^2-(m^2-m+1)+x=0

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

LHS =
(m^2- (m^2-m+1))^2-(m^2-m+1)+m
=(cancel(m^2)- cancel(m^2)+m-1))^2-(m^2-m+1-m)

=(m-1))^2-(m-1)^2=0
the equation is satisfied.
Hence m is the answer

Mar 10, 2016

let's put

x=sqrt(7+sqrt(7- sqrt(7+sqrt(7-sqrt....

We can easily see that

sqrt(7+sqrt(7-x))=x

So let's solve the equation:

7+sqrt(7-x)=x^2

sqrt(7-x)=x^2-7

7-x=(x^2-7)^2=x^4-14x^2+49

x^4-14x^2+x +42=0

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.