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 under which the roots of the equation \( x^3 - px^2 + qx - r = 0 \) are 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 the roots**: According to Vieta's formulas, the sum of the roots is equal to \( p \):
\[
(a - d) + a + (a + d) = 3a = p \implies a = \frac{p}{3}
...