Quantitative Aptitude :: progressions - Discussion

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Explanation:
Suppose the roots are r and s. Then (x-r)(x-s) = 0, so
=> x^2 - (r+s) x + (rs) = 0
and hence r+s = -b/a and rs = c/a.
We are saying that
=> r + s = (1/r^2 + 1/s^2)
=> (rs)^2 (r+s) = (rs)^2 (1/r^2 + 1/s^2) = s^2 + r^2
=> (rs)^2 (r+s) = (r+s)^2 - 2rs
=> (c/a)^2 (-b/a) = (-b/a)^2 - 2(c/a)
Multiplying through by a^3 we get
-bc^2 = ab^2 - 2(a^2)c
Divide through by abc to get
-c/a = b/c - 2a/b
or a/b - c/a = b/c - a/b
which means c/a, a/b and b/c are in AP.
Hence their reciprocals, a/c, b/a and c/b are in harmonic progression (HP).