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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