If a1- a2- a3-an are in H-P- then a1-a2-a3-an-a2-a1-a3-an-an-a1-a2-an-1 are in

The correct option is C H.PGiven a _ 1 , a _ 2 , a _ 3 ,..., a _ n are in H.P. ⇒ 1 / a _ 1 , 1 / a _ 2 , 1 / a _ 3 ,...., 1 / a _ n are in A.P ...

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Byju's Answer

Standard XII

Mathematics

Harmonic Progression

If a1, a2, ...

Question

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

A.P

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G.P

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H.P

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A.G.P

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Solution

The correct option is C H.P
Given $a_{1}, a_{2}, a_{3}, . . ., a_{n}$ are in H.P.
$\Rightarrow \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, . . . ., \frac{1}{a_{n}}$ are in A.P
$\Rightarrow \frac{a_{1} + a_{2} + a_{3} + . . . . + a_{n}}{a_{1}}, \frac{a_{1} + a_{2} + a_{3} + . . . . + a_{n}}{a_{2}}, \frac{a_{1} + a_{2} + a_{3} + . . . . + a_{n}}{a_{3}}, . . . ., \frac{a_{1} + a_{2} + a_{3} + . . . . + a_{n}}{a_{n}}$ are in A.P
Since we multiply fix quantity in A.P still it remains A.P.
$\Rightarrow \frac{a_{1} + (a_{2} + a_{3} + . . . . + a_{n})}{a_{1}}, \frac{{a_{2} + (a}_{1} + a_{3} + . . . . + a_{n})}{a_{2}}, \frac{{a_{3} + (a}_{1} + a_{2} + . . . . + a_{n})}{a_{3}}, . . . ., \frac{a_{n} + (a_{1} + a_{2} + a_{3} + . . . . + a_{n - 1})}{a_{n}}$ are in A.P
$\Rightarrow 1 + \frac{(a_{2} + a_{3} + . . . . + a_{n})}{a_{1}}, 1 + \frac{{(a}_{1} + a_{3} + . . . . + a_{n})}{a_{2}}, 1 + \frac{{(a}_{1} + a_{2} + . . . . + a_{n})}{a_{3}}, . . . ., 1 + \frac{(a_{1} + a_{2} + a_{3} + . . . . + a_{n - 1})}{a_{n}}$ are in A.P
$\Rightarrow 1 + {\frac{(a_{2} + a_{3} + . . . . + a_{n})}{a_{1}}, \frac{{(a}_{1} + a_{3} + . . . . + a_{n})}{a_{2}}, \frac{{(a}_{1} + a_{2} + . . . . + a_{n})}{a_{3}}, . . . ., \frac{(a_{1} + a_{2} + a_{3} + . . . . + a_{n - 1})}{a_{n}}}$ are in A.P
$\Rightarrow \frac{(a_{2} + a_{3} + . . . . + a_{n})}{a_{1}}, \frac{{(a}_{1} + a_{3} + . . . . + a_{n})}{a_{2}}, \frac{{(a}_{1} + a_{2} + . . . . + a_{n})}{a_{3}}, . . . ., \frac{(a_{1} + a_{2} + a_{3} + . . . . + a_{n - 1})}{a_{n}}$ are in A.P.
$\Rightarrow \frac{a_{1}}{a_{2} + a_{3} + . . . . + a_{n}}, \frac{a_{2}}{a_{1} + a_{3} + . . . . + a_{n}}, \frac{a_{3}}{a_{1} + a_{2} + . . . . + a_{n}}, . . . ., \frac{a_{n}}{a_{1} + a_{2} + . . . . + a_{n - 1}}$ are in H.P.

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