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Progressions

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CAT 2025 Lesson : Progressions - HP Concepts

A sequence

x_1,x_2,x_3,....,x_n

is said to be in Harmonic Progression if

\dfrac{1}{x_1},\dfrac{1}{x_2},\dfrac{1}{x_3},.....,\dfrac{1}{x_n}

are in Arithmetic Progression.

1) If

a

b

c

are in HP, then

\dfrac{1}{b} - \dfrac{1}{a} = \dfrac{1}{c} - \dfrac{1}{b}

and

b = \dfrac{2ac}{a + c}

2) In an HP, the middle term is the Harmonic Mean.

If the

2^{nd}

and

3^{rd}

terms of an HP are

9

and

6

respectively, then what is the

6^{th}

term?

As HP is the reciprocal of terms in an AP, let the terms in this HP be

\dfrac{1}{a},\dfrac{1}{(a + d)},\dfrac{1}{(a + 2d)},......

2^{nd}

term =

\dfrac{1}{(a + d)} = 9

⇒

a + d = \dfrac{1}{9}

-----(1)

3^{rd}

term =

\dfrac{1}{(a + 2d)} = 6

⇒

a + 2d = \dfrac{1}{6}

-----(2)

Subtracting

(1)

from

(2)

⇒

d = \dfrac{1}{6} - \dfrac{1}{9} = \dfrac{1}{18}

Substituting in

(1)

⇒

a = \dfrac{1}{9} - \dfrac{1}{18} = \dfrac{1}{18}

6^{th}

term of HP =

\dfrac{1}{(a + 5d)} = \dfrac{1}{\left( \dfrac{1}{18} + \dfrac{5}{18} \right)}=\dfrac{18}{6}=3

Answer:

3

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