Question

If first and $(2n-1)$ ${}^{th}$ terms of an A.P., G.P. and H.P. are equal and their $n$ ${}^{th}$ terms are $a,b,c$ respectively, then _____________

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

Answer 1

Answer: (C), (D)

$a>b>c$
$ac-b^2=0$

Let the first and the $(2n-1)$term be $\alpha$ and $\beta$ respectively.
Now, in the A.P., the term a is equidistant from $\alpha$ and $\beta$.
Thus, $a=\frac{\alpha +\beta }{2}=A.M.$
Now similarly, $b$ is equidistant from $\alpha ,\beta$ in the G.P.
So,$b=\sqrt{\alpha \beta }=GM$
Similarly, $c$ will be HM of $\alpha ,\beta$.

$c=\frac{2\alpha \beta }{\alpha +\beta }$

Since AM $>$ GM $>$ HM, $a>b>c$.
Also, it can be easily verified that $b^2=ac$.