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Question

If a+1a=p and a0; then show that:
a3+1a3=p(p23)

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Solution

a+1a=p
Squaring both sides
(a+1a)2=p2
a2+2(a)1a+(1a)2=p2
a2+2+1a2=p2
a2+1a2=p22
Now,
a3+1a3=(a+1a)(a2a(1a)+(1a2))
a3+1a3=(p)(p221)
a3+1a3=(p)(p23)
L.H.S=R.H.S

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