Question

If three real numbers $\mathrm{a},\mathrm{b}$ and $\mathrm{c}$ are in harmonic progression, then which of the following is true?

• A. $\frac{1}{\mathrm{a}},\mathrm{b},\frac{1}{\mathrm{c}}$ are in AP
• B. $\frac{1}{\mathrm{bc}},\frac{1}{\mathrm{ca}},\frac{1}{\mathrm{ab}}$ in HP
• C. $\mathrm{ab},\mathrm{bc},\mathrm{ac}$ are in HP
• D. $\frac{\mathrm{a}}{\mathrm{c}},\frac{\mathrm{b}}{\mathrm{c}},\frac{\mathrm{c}}{\mathrm{a}}$ are in HP

Answer 1

The correct option is B

$\frac{1}{\mathrm{bc}},\frac{1}{\mathrm{ca}},\frac{1}{\mathrm{ab}}$ in HP

Explanation for the correct option:

Finding the relation:

Since, $\mathrm{a},\mathrm{b},\mathrm{c}$ are in H.P. then the reciprocal of terms are in A.P.

$\therefore \frac{1}{\mathrm{a}},\frac{1}{\mathrm{b}},\frac{1}{\mathrm{c}}$are in A.P.

Multiply by $\mathrm{abc}$.

$\Rightarrow \frac{\mathrm{abc}}{\mathrm{a}},\frac{\mathrm{abc}}{\mathrm{b}},\frac{\mathrm{abc}}{\mathrm{c}}$ are in A.P.

$\Rightarrow \mathrm{bc},\mathrm{ca},\mathrm{ab}$ are in A.P.

$\Rightarrow \frac{1}{\mathrm{bc}},\frac{1}{\mathrm{ca}},\frac{1}{\mathrm{ab}}$ are in H.P.

Hence, the correct option is B.