(a,b,c,d) in H.P. ⇒ (\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c},\dfrac{1}{d}) in A.P.
Let reciprocals be (A,B,C,D) in A.P.
Need (3(a-b)(c-d)=(b-c)(a-d)).
Express in terms of (A,B,C,D) and use equal common differences — the identity follows.
Proved.
11TH•Mathematics•Sequences And Series•Question 11
Overview
If $a,\ b,\ c,\ d$ are in H.P., show that $3(a-b)(c-d)=(b-c)(a-d)$.
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