If H₁,H₂,H₃,………Hn be n harmonic means between a and b then$\frac{H_1+a}{H_1-a}+\frac{H_n+b}{H_n-b}=$

$$\begin{gathered} \mathrm{a},{\mathrm{H}}_1,{\mathrm{H}}_2,{\mathrm{H}}_3,\dots \dots .{\mathrm{H}}_{\mathrm{n}},\mathrm{b} \mathrm{areinH}.\mathrm{P} \\ \frac{1}{\mathrm{a}},\frac{1}{{\mathrm{H}}_1},\frac{1}{{\mathrm{H}}_2},\frac{1}{{\mathrm{H}}_3},\dots \dots ..\frac{1}{{\mathrm{H}}_{\mathrm{n}}},\frac{1}{\mathrm{b}} \mathrm{areA}.\mathrm{P} \end{gathered}$$
$$\begin{gathered} \begin{array}{l}\frac{1}{{\mathrm{H}}_1}=\frac{1}{\mathrm{a}}+\mathrm{d},\frac{1}{{\mathrm{H}}_{\mathrm{n}}}=\frac{1}{\mathrm{b}}-\mathrm{d} \\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{{\mathrm{H}}_1+\mathrm{a}}{{\mathrm{H}}_1-\mathrm{a}}+\frac{{\mathrm{H}}_{\mathrm{n}}+\mathrm{b}}{{\mathrm{H}}_{\mathrm{n}}-\mathrm{b}}=\frac{\frac{1}{\mathrm{a}}+\frac{1}{{\mathrm{H}}_1}}{\frac{1}{\mathrm{a}}-\frac{1}{{\mathrm{H}}_1}}+\frac{\frac{1}{\mathrm{b}}+\frac{1}{{\mathrm{H}}_{\mathrm{n}}}}{\frac{1}{\mathrm{b}}-\frac{1}{{\mathrm{H}}_{\mathrm{n}}}}\\ =\frac{\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{a}}+\mathrm{d}}{\frac{1}{\mathrm{a}}-\frac{1}{\mathrm{a}}-\mathrm{d}}+\frac{\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{b}}-\mathrm{d}}{\frac{1}{\mathrm{b}}-\frac{1}{\mathrm{b}}+\mathrm{d}} \\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{\frac{2}{\mathrm{a}}+\mathrm{d}}{-\mathrm{d}}+\frac{\frac{2}{\mathrm{b}}-\mathrm{d}}{\mathrm{d}}=\frac{\frac{2}{\mathrm{b}}-\mathrm{d}-\frac{2}{\mathrm{a}}-\mathrm{d}}{\mathrm{d}}\end{array} \end{gathered}$$
$$\begin{gathered} \begin{array}{l}=\frac{\frac{2}{\mathrm{b}}-\frac{2}{\mathrm{a}}-2\mathrm{d}}{\mathrm{d}}=\frac{\frac{2}{\mathrm{b}}-\frac{2}{\mathrm{a}}}{\mathrm{d}}-2 \\ =\frac{\frac{2}{\mathrm{a}}+2(\mathrm{n}+1)\mathrm{d}-\frac{2}{\mathrm{a}}}{\mathrm{d}}-2 \\ \left(\therefore \frac{1}{\mathrm{b}}=\frac{1}{\mathrm{a}}+(\mathrm{n}+2-1)\mathrm{d}=\frac{1}{\mathrm{a}}+(\mathrm{n}+1)\mathrm{d}\right)\\ =\frac{2\mathrm{d}(\mathrm{n}+1)}{\mathrm{d}}-2=2\mathrm{n}+2-2=2\mathrm{n}\end{array} \end{gathered}$$