Find the conditions if roots of the equation `x^(3)-px^(2)+qx-r=0` are in
(i) AP (ii)GP
(iii) HP

Explanation

To find the conditions for the roots of the equation \( x^3 - px^2 + qx - r = 0 \) to be in arithmetic progression (AP), geometric progression (GP), and harmonic progression (HP), we will analyze each case step by step.

### (i) Roots in Arithmetic Progression (AP)

1. **Assume the Roots**: Let the roots be \( a - d, a, a + d \).
2. **Sum of Roots**: By Vieta's formulas, the sum of the roots is given by:
\[
(a - d) + a + (a + d) = 3a = p
...