“Here’s a concise step-by-step solution:

1. Arithmetic Progression (AP): Since a, b, c are in AP, the following relationship holds true: 2b = a + c

2. Harmonic Progression (HP): Since a^2, b^2, c^2 are in HP, their reciprocals, \frac{1}{a^2}, \frac{1}{b^2}, \frac{1}{c^2}, are in AP. This gives us the relationship: \frac{2}{b^2} = \frac{1}{a^2} + \frac{1}{c^2}

3. Combine the equations: From the HP condition, we can rewrite the equation as: \frac{2}{b^2} = \frac{a^2 + c^2}{a^2c^2} 2a^2c^2 = b^2(a^2 + c^2)

4. Substitute the AP condition into the combined equation: From the AP condition, we know b = \frac{a+c}{2}. Substitute this into the equation from step 3: 2a^2c^2 = \left(\frac{a+c}{2}\right)^2(a^2 + c^2) 2a^2c^2 = \frac{(a+c)^2}{4}(a^2 + c^2) 8a^2c^2 = (a+c)^2(a^2 + c^2)

5. Expand and simplify: 8a^2c^2 = (a^2 + 2ac + c^2)(a^2 + c^2)

6. Consider possible solutions:

• If a = c, then from the AP condition, 2b = 2a, which means b = a. Therefore, a = b = c is a valid solution.

• The equation simplifies to a^4 + 2a^3c – 6a^2c^2 + 2ac^3 + c^4 = 0. One solution to this equation is a = c. If a, b, c are non-zero, the condition a=b=c satisfies both AP and HP conditions (a^2, a^2, a^2 are in HP).

The relationship between a, b, and c is most likely a=b=c, or one of them is zero leading to the others being zero.”