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Question

If xa+yb=1 and x2a2+y2b2=aba+b then prove that xn+1a+yn+1b=(aba+b)n.

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Solution

Solving the first two equations for x and y we have by eliminating y,
[(a+b)xab]2=0
x=aba+b=y.
xn+1a+yn+1b
=(aba+b)n+1[1a+1b]
=(aba+b)n+1[a+bab]=(aba+b)n

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