# Fibonnaci Nested Radicals

**Algebra**Level 5

For the given nested radical, a real number L exists as the evaluation of the expression.

\(2\cdot 2\sqrt { 2\sqrt [ 3 ]{ 2\sqrt [ 5 ]{ 2... } } } \) = L

If the sum of the first \(\left\lceil L \right\rceil \) terms of the decimal expansion of L is S, then what is S?

Assumptions:

The nth radical index is the nth fibonacci number. The first two numbers can be written as numbers with a radical of index 1.

Inspired by Brian Charlesworth's problem Harder Nested Radicals