SOLUTION: If a,b,c,d are in H.P., prove that ab+bc+cd = 3ad

Then the reciprocals  are in AP.

Let 2k = the common difference of the AP
Let m = the mean of the four terms of the AP 

[Note: m is the number half-way between the middle two terms, so 1/b is k 
less than m and 1/a is 2k less than 1/b or 3k less than m.  Similarly 1/c
is k more than m and 1/d is 2k more than 1/c or 3k more than m.)

So , , , 

and , , , 



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Edwin