The \( \infty \)-radical madness

Algebra Level 5

Let \[R_1=\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2-...}}}}} \] be a 5-periodic nested radical, with respective sign pattern is \( (++---) \).

Let \(R_2, R_3, R_4, R_5\) be analogous radicals with sign patterns \( (-++--) \), \( (--++-) \), \( (---++) \), \( (+---+) \).