1.

Insert 4harmonic means between 1&1/6

Answer»

Let a1, a2, a3, a4 are required harmonic means.

Then 1, a1, a2, a3, a4, \(\frac16\) is a harmonic sequence.

∴ \(1, \frac1{a_1}, \frac1{a_2},\frac1{a_3},\frac1{a_4},6\) is an arithmetic sequence.

∴ \(d = \frac{b-a}{n +1} = \frac{6 - 1}{4 + 1}=\frac55 = 1\)

∴ \(\frac1{a_1} = a + d = 1 + 1 = 2\)

⇒ \(a_1 = \frac12\)

\(\frac1{a_2} = a + 2d = 1 + 2 = 3\)

⇒ \(a_2 = \frac13\)

\(\frac1{a_3} = a + 3d = 1 + 3 = 4\)

⇒ \(a_3 = \frac14\) 

\(\frac1{a_4} = a + 4d = 1 +4 = 5\)

⇒ \(a_4= \frac15\)

Hence, \(\frac12, \frac13,\frac14\) & \(\frac15\) are inserted hormonic means between 1 & \(\frac16\).


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