First slide

Harmonic Progression

Question

$If a_{1}, a_{2}, a_{3}, \dots \dots a_{n} are in H.P then a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + \dots + a_{n - 1} a_{n} =$

Moderate

A

$(n - 1) a_{1} a_{n}$
B

$(n + 1) a_{1} a_{n}$
C

$(n - 2) a_{1} a_{n}$
D

$(n + 4) a_{1} a_{n}$

Solution

$\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}} ..... \frac{1}{a_{n}} a r e i n A P$
$\begin{array}{l} ∴ \frac{1}{a_{2}} - \frac{1}{a_{1}} = \frac{1}{a_{3}} - \frac{1}{a_{2}} = \dots = \frac{1}{a_{n}} - \frac{1}{a_{n - 1}} = d \\ \Rightarrow \frac{a_{1} - a_{2}}{a_{1} a_{2}} = \frac{a_{2} - a_{2}}{a_{2} a_{3}} = \dots \cdot = \frac{a_{n - 1} - a_{n}}{a_{n - 1} a_{n}} = d \\ \Rightarrow \frac{a_{1} - a_{2} + a_{2} - a_{3} + \dots . . + a_{n - 1} - a_{n}}{a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}} = d \end{array}$
$\Rightarrow a_{1} a_{2} + a_{2} a_{3} + . . . . . + a_{n - 1} a_{n} = \frac{a_{1} - a_{n}}{d} = \frac{(n - 1) d a_{1} a_{n}}{d} = (n - 1) a_{1} a_{n}$

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