If a>0, b>0, c>0 are in G.P., then logax,logbx,logcx are in
H.P.
a, b, c are in G.P.
∴b2=ac
Let logax=n
∴x=an
Similarly,
x=bl and
x=cm
an=bl=cm=k(say)
⇒a=k1n b=k1l c=k1n
Substitute the values of a, b, c in b2=ac
2l=1n+1m
Hence 1n,1l,1m are in A.P. or n, l, m are in H.P.