Find the value of : $\lim_{n\to\infty}(2a)^{\frac{n}{2}}\sqrt{a-\sqrt{a(a-1)+\sqrt{a(a-1)+\cdots}}}$

Solution 1:

The answer given was completely wrong.

Solution 2:

Solution Outline (I will leave the details for you to fill).

So, we consider this: $$ (2a)^{\frac{n}{2}}\underbrace{\sqrt{a-\sqrt{a(a-1)+\sqrt{a(a-1)+\cdots}}}}_{n \textrm{ square roots}}, $$ where $n \to \infty$. We assert that the number of square roots that are nested is indeed just infinity. So, we consider this: $$ S = \sqrt{a(a-1)+\sqrt{a(a-1)+\cdots}}, $$ where, here, there are an infinite number of nested square roots. Then, $$S^2=a(a-1)+S.$$Then, you can solve for $S$ using the quadratic formula. our original expression is, remember the following: $$ (2a)^{\frac{n}{2}}\sqrt{a-S}. $$You can get $S$ using the quadratic formula and substitute into this expression. Then, finding the limit of the resulting expression will not be trivial but it won't be impossible either.