Random Math Problems 1- Computing Nested Infinite Radicals

Compute


This is a problem containing an infinitely nested radicals of 20,where you have to compute the entire nested radical.It goes on forever which is a hurdle for our intuitive understanding.

Mathematical Steps (Algorithm)

First we need to assign this entire infinitely nested radicals to a variable,like s ,and declare it as equation (1).

            

We need to square both sides of the equation to separate one integer out of the infinitely nested radical so that it becomes an rational number+ infinitely nested radical. We are going to need this rational number later. We have to declare it as equation (2).

               

now we need to subtract equation (1) from equation (2)                 

              

                                 Now we notice that the infinitely nested radicals cancel out.

                      (we can factor out s )

                  

Now we look for two consecutive integers among the factors of 20 which can satisfy this equation

            Factors of 20 = 1,2,4,5,10,20

4 and 5 are the consecutive integers. s has to the bigger one,as s>(s-1)

            Therefore, s=5

Statement: One argument can be made against assigning variables to infinity, because infinity is not a number, it’s a concept. One can solve questions like is 0.999.. = 1? , using variables and prove them equal, but this confusion is the reason why I think limits in calculus are introduced.

Saikanam Siam
Junior
Brooklyn Technical High School