1. **Problem statement:** Given that $a^p = b^q = c^r$ and $a, b, c$ are in geometric progression (G.P.), prove that $p, q, r$ are in harmonic progression (H.P.). 2. **Recall definitions:** - If $a, b, c$ are in G.P., then $b^2 = ac$. - If $p, q, r$ are in H.P., then their reciprocals $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ are in arithmetic progression (A.P.), i.e., $2/ q = 1/p + 1/r$. 3. **Given:** $$a^p = b^q = c^r = k \quad \text{(some constant)}$$ 4. **Express $a, b, c$ in terms of $k$ and $p, q, r$: ** $$a = k^{1/p}, \quad b = k^{1/q}, \quad c = k^{1/r}$$ 5. **Use the G.P. condition $b^2 = ac$: ** $$\left(k^{1/q}\right)^2 = k^{1/p} \times k^{1/r}$$ $$k^{2/q} = k^{1/p + 1/r}$$ 6. **Since $k > 0$ and $k \neq 1$, equate exponents:** $$\frac{2}{q} = \frac{1}{p} + \frac{1}{r}$$ 7. **Rewrite:** $$2 \times \frac{1}{q} = \frac{1}{p} + \frac{1}{r}$$ 8. **Interpretation:** This means $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ are in arithmetic progression. 9. **Conclusion:** Therefore, $p, q, r$ are in harmonic progression. **Final answer:** $p, q, r$ are in H.P.