If a1- a2- a3- -an are in H-P- then a1a2-a3-an- a2a1-a3-an- ana1-a2-an-1 are in

The correct option is C H.P. a_1, a_2, a_3, .....a_n are in H.P. 1/a_1, nbsp;1/a_2, nbsp;1/a_3 .....1/a_n are in A.P.Multiply each term by ...

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Byju's Answer

Standard XII

Mathematics

Harmonic Mean

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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None of these

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Solution

The correct option is C H.P.
$a_{1}, a_{2}, a_{3}, . . . . . a_{n}$ are in H.P. $\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}} . . . . . \frac{1}{a_{n}}$ are in A.P.
Multiply each term by $a_{1} + a_{2} + a_{3} + . . . . + a_{n}$ , then subtract $1$ from each term
We get $\frac{a_{2} + a_{3} + . . . . + a_{n}}{a_{1}}, \frac{a_{1} + a_{3} + . . . . + a_{n}}{a_{2}}, . . . . . \frac{a_{1} + a_{2} + . . . . + a_{n - 1}}{a_{n}}$ are in A.P.
$∴ \frac{a_{1}}{a_{2} + a_{3} + . . . . + a_{n}}, \frac{a_{2}}{a_{1} + a_{3} + . . . . + a_{n}}, . . . . \frac{a_{n}}{a_{1} + a_{2} + . . . . + a_{n - 1}}$ in H.P.

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If a1,a2,a3,.....an are in H.P., then a1a2+a3+...an,a2a1+a3+....+an,ana1+a2+...+an−1 are in

The correct option is C H.P.a1,a2,a3,.....an are in H.P. 1a1,1a2,1a3.....1an are in A.P.Multiply each term by a1+a2+a3+....+an, then subtract 1 from each termWe get a2+a3+....+ana1,a1+a3+....+ana2,.....a1+a2+....+an−1an are in A.P.∴a1a2+a3+....+an,a2a1+a3+....+an,....ana1+a2+....+an−1 in H.P.

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