Roots in Arithmetic, Geometric and Harmonic Progression

Sometimes a problem does not simply hand us the roots of a cubic -- it tells us the roots satisfy an extra structural condition, such as being in arithmetic, geometric, or harmonic progression, and asks us to find them using this together with the coefficient relations of §4.1.2.

Roots in A.P. are conveniently written as $a-d,\ a,\ a+d$, since their sum is simply $3a$. Setting $3a=S_1$ pins down $a$ immediately (a rational number in a well-posed problem); the value of $d$ is then found from the product relation $S_3=a(a^2-d^2)$ or from $S_2$.

Roots in G.P. are written as $\dfrac ar,\ a,\ ar$, since their product is simply $a^3$. Setting $a^3=S_3$ pins down $a$ (a perfect cube in a well-posed problem); the value of $r$ is then found, typically as the root of a quadratic, from the sum relation $S_1=a\big(\tfrac1r+1+r\big)$. …