series p/q Watch

psanghaLFC
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#1
Report Thread starter 10 years ago
#1
If P_1=2 and Q_1=1 and

p_(n+1)= (P_n)^2+3(Q_n)^2 Q_(n+1)=2p_n*Q_n

prove that


P_n/Q_n>sqrt(3)


how could i go about proving this? Could induction be usefull?
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DoMakeSayThink
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#2
Report 10 years ago
#2
Yes, indeed it could. Have you tried?
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psanghaLFC
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#3
Report Thread starter 10 years ago
#3
Ok, i think i may have done it but i didn't use induction

if
p_(n+1)/q_(n+1)>sqrt3

then


(p_(n+1)/q_(n+1))^2-3>0
so
(((p_n)^2+(3(q_n)^2)^2)/(4(p_(n)^2q_(n))-3>0


i will write p_n as pn now

(pn)^4+6(pn)^2(qn)^2+9(qn)^4-12(pn)^2(qn)^2>0

so

((pn)^2-3(qn))^2>0
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psanghaLFC
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#4
Report Thread starter 10 years ago
#4
I haven't refered to whether or not the series is strictley decreasing or increasing, i don't believe it fluctuates and if it was increasing there would be no point asking the question.
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