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  4. If the first and the (2n - 1)th term of an AP, GP and HP are equal and their nth terms are a, b and c respectively, then

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Q. If the first and the $ (2n - 1)th$ term of an $AP, GP$ and $HP$ are equal and their $n$th terms are $a, b$ and $c$ respectively, then

IIT JEEIIT JEE 1988Sequences and Series

Solution:

Since, first and $(2n - 1)$th terms are equal.
Let first term be $x$ and $(2n - 1)$th term be $y$,
whose middle term is $t_n.$
Thus, in arithmetic progression, $t_n =\frac{x+y}{2} = a$
In geometric progression, $t_n =\sqrt {xy} = b$
In harmonic progression, $t_n =\frac{2xy}{x+y} =c $
$ \Rightarrow b^2 = ac $ and $a > b >c$ [using AM > GM > HM]
Here, equality holds (i.e. $a = b =c$) only if all terms are
same. Hence, options (a), (b) and (d) are correct