Prove that a,b,c are in G.P., if $l o g_{a} n, l o g_{b} n, l o g_{c} n$ are in H.P.

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Solution

$l o g_{a} n, l o g_{b} n, l o g_{c} n$ are in H.P.
Taking reciprocals
$∴ l o g_{n} a, l o g_{n} b, l o g_{n} c$ are in H.P.
or $2 l o g_{n} b = l o g_{n} a + l o g_{n} c$
or $l o g_{n} b^{2} = l o g_{n} a c ∴ b^{2} = a c$
Hence a,b,c are in G.P.

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