Q13/036) if a,b,c,d are in harmonic progression then prove ab+bc+cd=3ad.

a b c d are in HP

so 1/a , 1/b. 1/c , 1/d are in AP

let common difference be m

m = 1/b - 1/a = (a-b)/ab

or ab = (a-b)/m

similarly
bc = ( b-c)/m

and
cd = (c-d)/m

adding we get ab + bc + cd = (a-d )/m ... 1

further 1/d - 1/a = 3m or (a-d) = 3mda ...2

for (1) and (2) we get ab + bc + cd = (a-d )/m = 3mad / m = 3ad

proved