# 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 $$(-++--)$$, $$(--++-)$$, $$(---++)$$, $$(+---+)$$.

Find $$R_1-R_5+R_2-R_4+R_3$$.

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