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Please provide the solution
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If the terms are in HP, the reciprocal of each term is in AP.

Thus we can transform the question into a simple AP question to solve.

The fifth term of the AP = \(\frac{3}{2}\)

The ninth term of the AP =\(\frac{9}{2}\)

The first term of the AP = x

\(\frac{3}{2}\) = x + (5-1)d

\(\frac{3}{2}\) = x + 4d ---------------- (1)

Similarly

\(\frac{9}{2}\) = x + (9-1)d

\(\frac{9}{2}\) = x + 8d ---------------- (2)

From both the equations d = 3/4

Therefore a =

\(\frac{3}{2}\) = x + 4\(\frac{3}{4}\)

x = -1.5

We know that x is the first term of AP,

the first term of HP = \(\frac{1}{x}\) = \(\frac{1}{(-1.5) }\)

= \(\frac{-10}{15}\)

= \(\frac{-2}{3}\)

IMO - C
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The 5th and 9th terms of a harmonic progression are 2/3 and 2/9 respectively.

What is the first term of the progression? (A harmonic progression is a sequence of numbers formed by taking the reciprocals of an arithmetic progression.)

Arithmetic progression:-
First term = a
Common difference =d
Fifth term = a+4d = 3/2 =1.5
Ninth term = a+8d =9/2 =4.5
4d = 4.5-1.5=3
d=.75
a= -1.5=-3/2

Corresponding first term of harmonic progression =1/a = -2/3=-.67

IMO C
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