Maths / Arithmetic Progression / Sum of First n Terms

QUESTION

If $\dpi{120} \fn_jvn \large x_1,x_2,x_3,--------x_n$ are in AP,then the value of $\dpi{120} \fn_jvn \large \frac{1}{x_1x_2}+\frac{1}{x_2x_3}+\frac{1}{x_3x_4}+-------\frac{1}{x_n-1x_n}$ is

 OPTIONS A. $\dpi{80} \fn_phv \large \frac{n-1}{x_1x_n}$ B. $\dpi{80} \fn_phv \large \frac{n-1}{x_2x_n-1}$ C. $\dpi{80} \fn_phv \large \frac{n}{x_1x_n}$ D. None of the above
Right Option : A

EXPLANATION
Explain TypeExplanation Content
Text

$\dpi{80} \fn_phv \large \frac{1}{x_1x_2}+\frac{1}{x_2x_3}+\frac{1}{x_3x_4}+----+\frac{1}{x_n-1x_n}$ is

$\dpi{80} \fn_phv \large \frac{1}{d}\left ( \frac{d}{x_1x_2}+\frac{d}{x_2x_3}+-----+\frac{d}{x_n-1x_n} \right )$

=$\dpi{80} \fn_phv \large \frac{1}{d}\left ( \frac{x_2-x_1}{x_2x_1}+\frac{x_3-x_2}{x_3x_2}+----\right )$$\dpi{80} \fn_phv \large +\frac{x_n-x_n-1}{x_n-1x_n}$

=$\dpi{80} \fn_phv \large \frac{1}{d}\left ( \frac{1}{x_1}-\frac{1}{x_2}+\frac{1}{x_2}-\frac{1}{x_3}+---- \right )$$\dpi{80} \fn_phv \large +\frac{1}{x_n-1}-\frac{1}{x_n}$

=$\dpi{80} \fn_phv \large \frac{1}{d}\left ( \frac{1}{x_1}-\frac{1}{x_n} \right )=\frac{1}{d}$$\dpi{80} \fn_phv \large \frac{x_n-x_1}{x_1x_n}=\frac{1}{d}[\frac{(n-1)d}{x_1x_n}]$

=$\dpi{80} \fn_phv \large \frac{(n-1)}{x_1x_n}$

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