If H_1, H_2, H_3........H_2n+1 are in H.P. then is equal

JEE MainMathematicsModerateSequences and Series

If H_1, H_2, H_3........H_2n+1 are in H.P. then
$\sum_{i = 1}^{2n}\mspace{2mu}( - 1)^{i}\left( \frac{H_{i} + H_{i + 1}}{H_{i} - H_{i + 1}} \right)$
is equal

Show answer and solution

Let

\frac{1}{H_{i + 1}} - \frac{1}{H_{i}} = k

\therefore\sum_{i = 1}^{2n}\mspace{2mu}( - 1)^{i}\left( \frac{H_{i} + H_{i + 1}}{H_{i} - H_{i + 1}} \right)

= \sum_{i = 1}^{2n}\mspace{2mu}\frac{( - 1)^{i}}{k}\left( \frac{1}{H_{i + 1}} + \frac{1}{H_{i}} \right) = 2n

Source: Grass-G / M-19 Mathematics Paper 28 Nov Fourth.docx

Scan any doubt from your notes and get a whiteboard explanation.