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Question

If pnqn be the nth convergent to a2+1, show that p22+p23+....+p2n+1q22+q23+....+q2n+1=pn+1pn+2p1p2qn+1qn+2q1q2.

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Solution

a2+1=a+12a+12a+12a+......
pn+1=2apn+1+pn
pn+1=2apn+pn1
pn=2apn+pn1
p3=2ap2+p1
Multiply these terms with pn+1,pn,.....p2 respectively, add and erase the terms
pn+1pn,pnpn1,.......p3p2 from each side of the sum; we obtain
pn+2.pn+1=2a(p2n+1+p2n+....p22)+p1p2
pn+2.pn+1p1p2=2a(p2n+1+p2n+....p22)
qn+2=2aqn+1+qn
qn+1=2aqn+qn1
qn=2aqn1+qn2
q3=2aq2+q1
Multiply these equations by qn+1,qn,.....q2 respectively, add and erase the terms,
qn+2.qn+1=2a(q2n+1+q2n+....+q22)+q1q2
qn+2.qn+1q1q2=2a(q2n+1+q2n+....+q22)
pn+1.pn+2p1p2qn+1.qn+2q1q2=p22+p23+......+p2n+1q22+q23+......+q2n+1

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