Answer

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**Hint:**Here, we know that, H.P is the opposite of A.P. then,

a, b, c are in H.P,

$\dfrac{2}{\text{b}}=\dfrac{1}{\text{a}}+\dfrac{1}{\text{c}}\Rightarrow \dfrac{2\,\text{ac}}{\text{a}+\text{c}}$ and check all option which will be true.

**Complete step by step solution:**Given,

a, b, c are in H.P.

⇒ $\dfrac{1}{\text{b}}-\dfrac{1}{\text{a}}=\dfrac{1}{\text{c}}-\dfrac{1}{\text{b}}$

⇒ $\dfrac{1}{\text{b}}+\dfrac{1}{\text{b}}=\dfrac{1}{\text{a}}+\dfrac{1}{\text{c}}$

⇒ After addition, we get

⇒ $\dfrac{2}{\text{b}}=\dfrac{\text{a}+\text{c}}{\text{ac}}$

After cross-multiplication, we get,

⇒ $\dfrac{\text{b}}{2}=\dfrac{\text{ac}}{\text{a}+\text{c}}$

⇒ $\text{b}=\dfrac{\text{2 ac}}{\text{a}+\text{c}}$

Now, option (1)

⇒ $\dfrac{1}{\text{b}-\text{a}}+\dfrac{1}{\text{b}-\text{c}}=\dfrac{1}{\text{b}}$

⇒ After adding, we get

⇒ $\dfrac{\text{b}-\text{c}+\text{b}-\text{a}}{\text{(b}-\text{c)(b}-\text{a)}}=\dfrac{1}{\text{b}}$

\[\Rightarrow (2\text{b}-\text{a}-\text{c})\,\text{b}={{\text{b}}^{\text{2}}}\,-\,\text{ab}\,-\,\text{cb}\,+\,\text{ac}\]

\[\Rightarrow 2{{\text{b}}^{2}}-\text{ab}-\text{bc}={{\text{b}}^{\text{2}}}-\text{ab}-{{\text{b}}^{2}}\text{c}+\text{ac}\]

\[\Rightarrow {{\text{b}}^{2}}=\text{ac}\]

That is wrong.

Now, option (2)

$\Rightarrow \dfrac{\text{ac}}{\text{a}+\text{c}}=\text{b}$

That is wrong,

Now, option (3).

$\Rightarrow \dfrac{\text{b}+\text{a}}{\text{b}-\text{a}}+\dfrac{\text{b}+\text{c}}{\text{b}-\text{c}}=1$.

⇒ After cross-multiplication, we get,

\[\Rightarrow (\text{b}+\text{a})(\text{b}-\text{c})+(\text{b}+\text{c})(\text{b}-\text{a})=(\text{b}-\text{a})(\text{b}-\text{c})\]

Now, multiplying

\[\Rightarrow {{\text{b}}^{\text{2}}}-\text{bc}+\text{ab}-\text{ac}+{{\text{b}}^{\text{2}}}-\text{ab}+\text{bc}-\text{ac}={{\text{b}}^{\text{2}}}-\text{ab}-\text{bc}+\text{ac}\]

\[\Rightarrow {{\text{b}}^{\text{2}}}-2\text{ac}=-\text{ab}-\text{bc}+\text{ac}\]

\[\Rightarrow {{\text{b}}^{\text{2}}}=\dfrac{-\text{ab}-\text{bc}+\text{ac}}{2\text{ac}}\]

That is False.

Hence, the correct answer is none of the above.

**Note:**The above question is of arithmetic progression in which A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always same and the h.p is a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.

Here, H.P = $\dfrac{1}{\text{b}}-\dfrac{1}{\text{a}}=\dfrac{1}{\text{c}}-\dfrac{1}{\text{b}}$

,where, a,b.c and d is the number of a series.then, check all option.

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