# The $\infty$-Radical Madness 3

Calculus Level 5

Let $R_1=\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+...}}}}}$ $R_2=\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-...}}}}}$ $R_3=\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+...}}}}}$ $R_4=\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}-...}}}}}.$

Given that the values of the four infinitely nested radical expressions above can be expressed

$R_1=\frac{p_1}{q}, R_2=\frac{p_2}{q}, R_3=\frac{p_3}{q}, R_4=\frac{p_4}{q}$

where $p_1, p_2,p_3,p_4, q$ are positive integers with $\gcd(p_i,q) = 1$ for $1\leq i\leq 4$, find $p_1+p_2+p_3+p_4.$

Clarification: as a sequence, $R_4$ is defined in the following way: $R_4^1:=\sqrt{\frac{21}{16}}, R_4^n:=\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+R_4^{n-1}}},$ where $R_4^n$ is $n$-th term of the sequence. $R_4 := \lim_{n\to\infty} R_4^n.$

$R_3$ is defined analogously.

Warning: in $R_4$ you may need to extract square root from a negative number. This root is always principal (i.e $\sqrt{-2}=i\sqrt{2}$, not $-i\sqrt{2}$ ).

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