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Math Help - A question on miscellaneous series

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    Unhappy A question on miscellaneous series

    \frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz} and p,q,r be in AP , then prove that x,y,z are in HP.
    Last edited by CaptainBlack; August 31st 2009 at 08:39 PM.
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    Quote Originally Posted by matsci0000 View Post
    \frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz} and p,q,r be in AP , then prove that x,y,z are in HP.
    If \frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz} = c, then a = x(cp+1) = y(cq+1) = z(cr+1), from which \tfrac1x = \tfrac1a(cp+1), \tfrac1y = \tfrac1a(cq+1), \tfrac1z = \tfrac1a(cr+1). But if p, q and r are in AP then so are \tfrac1a(cp+1), \tfrac1a(cq+1) and \tfrac1a(cr+1).
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