If l, m, n are three numbers in G.P. then the ratio of first term to the common difference of an A.P. whoselth, mth and nth terms are in HP is equal to
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If x,y,z are in H.P. then
z
x
=
y−z
x−y
l,m,n are in G.P. Therefore
m−n
l−m
=
m
l
....(1)
We have a+(l−1)d,a+(m−1)d,a+(n−1)d in H.P.
Therefore,
a+(n−1)d
a+(l−1)d
=
(m−n)d
(l−m)d
=
m−n
l−m
=
m
l
....[From (1)]
⇒a(l−m)=d(l(m−n)+(l−m))
⇒
d
a
=
l−m
l(m−n)
+1=m+1
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