The \infty -Radical Madness 3

Calculus Level 5

Let R1=2116+2116+2116+2116+2116+...R_1=\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}+...}}}}} R2=21162116211621162116...R_2=\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}-...}}}}} R3=2116+21162116+21162116+...R_3=\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+...}}}}} R4=21162116+21162116+2116....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

R1=p1q,R2=p2q,R3=p3q,R4=p4q 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 p1,p2,p3,p4,qp_1, p_2,p_3,p_4, q are positive integers with gcd(pi,q)=1\gcd(p_i,q) = 1 for 1i41\leq i\leq 4, find p1+p2+p3+p4. p_1+p_2+p_3+p_4.

Clarification: as a sequence, R4 R_4 is defined in the following way: R41:=2116,R4n:=21162116+R4n1, R_4^1:=\sqrt{\frac{21}{16}}, R_4^n:=\sqrt{\frac{21}{16}-\sqrt{\frac{21}{16}+R_4^{n-1}}}, where R4n R_4^n is nn-th term of the sequence. R4:=limnR4n.R_4 := \lim_{n\to\infty} R_4^n.

R3 R_3 is defined analogously.

Warning: in R4 R_4 you may need to extract square root from a negative number. This root is always principal (i.e 2=i2 \sqrt{-2}=i\sqrt{2} , not i2 -i\sqrt{2} ).

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