Animated Solution for Mathematics - Sequence and Series: If $a,b,c$ are in A.P., $a^2,b^2,c^2$ are in H.P., then prove that either $a=b=c$ or $a,b,-c/2$ form a G.P.

We are given that $a,b,c$ are in Arithmetic Progression (A.P.). This implies that the common difference is constant, leading to the fundamental relation:

Next, we are given that $a^2,b^2,c^2$ are in Harmonic Progression (H.P.). By the definition of an H.P., the reciprocal of these terms must form an A.P., which leads to the harmonic mean property:

To bridge these two conditions, we substitute $b=\frac{a+c}{2}$ into the H.P. equation. This creates a unified equation:

Expanding the left side, we obtain:

To manage the complexity, let $X=a^2+c^2$. Note that $(a+c{)}^2=a^2+c^2+2ac=X+2ac$. Substituting these into the equation yields:

Factoring this quadratic equation, we find:

This gives us two distinct paths for the relationship between $a$ and $c$:

1. If $X=2ac$, then $a^2+c^2=2ac$, which simplifies to $(a-c{)}^2=0$. This implies $a=c$, resulting in a constant sequence where $a=b=c$.

2. If $X=-4ac$, then $a^2+c^2=-4ac$. Substituting back into the expression for $(a+c{)}^2$, we get $(a+c{)}^2=a^2+c^2+2ac=-4ac+2ac=-2ac$.

This leads to the condition $4b^2=-2ac$, or $b^2=a(-\frac{c}{2})$. This demonstrates the hidden constraints that force these sequences to maintain their specific mathematical symmetry.