If a_1, a_2, a_3, dots, a_n form a harmonic p | vedclass

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(D) Given that $a_1, a_2, a_3, \dots, a_n$ are in harmonic progression $(HP)$.Therefore,$\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots, \frac{1}{

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$(n - 1)a_1a_n$

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(D) Given that $a_1, a_2, a_3, \dots, a_n$ are in harmonic progression $(HP)$.Therefore,$\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots, \frac{1}{a_n}$ are in arithmetic progression $(AP)$.Let the common difference be $d = \frac{1}{a_{k+1}} - \frac{1}{a_k}$.Then,$a_k a_{k+1} = \frac{a_{k+1} - a_k}{d} = \frac{1}{d} (a_k - a_{k+1})$.Sum $S = \sum_{k=1}^{n-1} a_k a_{k+1} = \frac{1}{d} \sum_{k=1}^{n-1} (a_k - a_{k+1}) = \frac{1}{d} (a_1 - a_n)$.In the $AP$,$\frac{1}{a_n} = \frac{1}{a_1} + (n - 1)d$,so $d = \frac{a_1 - a_n}{(n - 1)a_1 a_n}$.Substituting $d$ into the sum expression:$S = \frac{a_1 - a_n}{\frac{a_1 - a_n}{(n - 1)a_1 a_n}} = (n - 1)a_1 a_n$.

If $a_1, a_2, a_3, \dots, a_n$ form a harmonic progression,find the value of $a_1a_2 + a_2a_3 + \dots + a_{n-1}a_n$.

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