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

The correct option is A H.P. 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. Multiply each term by a_1 + a_2 ...

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Byju's Answer

Standard XI

Mathematics

Harmonic Progression

If a1, a2, a3...

Question

If a₁, a₂, a₃.....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.

$a_{1}$ , $a_{2}$ , $a_{3}$ ........... $a_{n}$ 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 substract 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.

∴ $\frac{a_{1}}{a_{2} + a_{3} + . . . . . . . . + a_{n}}$ , $\frac{a_{2}}{a_{1} + a_{3} + . . . . . . . . . . . + a_{n}}$ , ........... $\frac{a_{n}}{a_{1} + a_{2} + . . . . . . . . + a_{n - 1}}$ are 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

If a₁, a₂, a₃.....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