Insert three harmonic means between 1-12 and 1-20-

The correct option is A 1/14 , 1/16 and 1/18 .After inserting the harmonic means, #160;let the harmonic progression be 1/a , #160; ...

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Standard XII

Mathematics

Nth Term of GP

Insert three ...

Question

Insert three harmonic means between $\frac{1}{12}$ and $\frac{1}{20}$ .

\frac{1}{14}

\frac{1}{16}

and

\frac{1}{18}

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\frac{1}{4}

\frac{1}{16}

and

\frac{1}{18}

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\frac{1}{14}

\frac{1}{6}

and

\frac{1}{18}

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\frac{1}{14}

\frac{1}{16}

and

\frac{1}{8}

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Solution

The correct option is A $\frac{1}{14}$ , $\frac{1}{16}$ and $\frac{1}{18}$ .
After inserting the harmonic means, let the harmonic progression be
$\frac{1}{a}$ , $\frac{1}{a + d}$ , $\frac{1}{a + 2 d}$ , $\frac{1}{a + 3 d}$ , $\frac{1}{a + 4 d}$
Given $\frac{1}{a} = \frac{1}{12}$ and $\frac{1}{a + 4 d} = \frac{1}{20} ⟹ a = 12$ and $d = 2$
$∴$ The required harmonic means are
$\frac{1}{14}$ , $\frac{1}{16}$ and $\frac{1}{18}$ .

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Insert three harmonic means between 112 and 120.

The correct option is A 114, 116 and 118.After inserting the harmonic means, let the harmonic progression be1a, 1a+d, 1a+2d, 1a+3d, 1a+4dGiven 1a=112 and 1a+4d=120⟹a=12 and d=2∴ The required harmonic means are114, 116 and 118.

Insert three harmonic means between $\frac{1}{12}$ and $\frac{1}{20}$ .