First slide

Harmonic Progression

Question

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

Moderate

A

${n(a}_{1} {-a}_{n})$
B

${(n-1)(a}_{1} {-a}_{n})$
C

${na}_{1} a_{n}$
D

${(n-1)a}_{1} a_{n}$

Solution

$\frac{1}{a_{2}} - \frac{1}{a_{1}} = \frac{1}{a_{3}} - \frac{1}{a_{2}} = .. = \frac{1}{a_{n}} - \frac{1}{a_{n-1}} = d$ $(\sin c e \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, . . . . . . . \frac{1}{a_{n}} a r e i n A . P)$
$\begin{array}{l} a_{1} a_{2} = \frac{a_{1} - a_{2}}{d}, a_{2} a_{3} = \frac{a_{2} - a_{3}}{d}, . . . ., a_{n - 1} a_{n} = \frac{a_{n - 1} - a_{n}}{d} \\ \Rightarrow g i v e n e x p r e s s i o n = \frac{1}{d} [a_{1} - a_{2} + a_{2} - a_{3} + \dots + a_{n - 1} - a_{n}] = \frac{a_{1} - a_{n}}{d} \\ b u t n t h t e r m = \frac{1}{a_{n}} = \frac{1}{a_{1}} + (n - 1) d \\ \Rightarrow \frac{a_{1} - a_{n}}{a_{1} a_{n}} = (n - 1) d \\ \Rightarrow \frac{a_{1} - a_{n}}{d} = (n - 1) a_{1} a_{n} \\ \Rightarrow g i v e n e x p r e s s i o n = (n - 1) a_{1} a_{n} \end{array}$

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