Question

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

• A. $a = b = c$
• B. $a \ge b \ge c$
• C. $a + c = b$
• D. $ac - b^2 =0$

Answer 1

Answer: (D)

$ac - b^2 =0$

Answer 2

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