Question 461296
I'm guessing "h.p." means harmonic progression? I recommend you do not use unusual abbreviations.


If a, b, c is a harmonic progression, then we can say that a = 1/k, b = 1/(k+d) and c = 1/(k+2d). We can replace the expressions for a, b, c into the next three terms:


*[tex \LARGE \frac{b+c}{bc} = \frac{ \frac{1}{k+d} \frac{1}{k+2d} }{\frac{1}{k+d} + \frac{1}{k+2d}} = \frac{1}{2k + 3d}]


Similarly,


*[tex \LARGE \frac{c+a}{ca} = \frac{ \frac{1}{k+2d} \frac{1}{k} }{\frac{1}{k+2d} + \frac{1}{k}} = \frac{1}{2k + 2d}]


*[tex \LARGE \frac{a+b}{ab} = \frac{ \frac{1}{k} \frac{1}{k+d} }{\frac{1}{k} + \frac{1}{k+d}} = \frac{1}{2k + d}]


We see the third, second, and first terms (in that order) follow a harmonic progression because the reciprocals of each make an arithmetic progression with common difference d.