Question

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

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

Answer 1

Answer: (A), (B), (D)

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