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Prove that any sequence of numbers a(1),...

Prove that any sequence of numbers `a_(1),a_(2),...a_(n)` satisfying the condition,
`(1)/(a_(1)a_(2))+(1)/(a_(1)a_(2))+...+(1)/(a_(n-1)a_(n))=(n-1)/(a_(1)a_(n))`
for every `n ge3`, is in arithmetic progression.

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