Question

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

• A. $a>b>c$
• B. $ac=b^2$
• C. $a+b=c$
• D. none of these

Answer 1

Answer: (B)

$ac=b^2$

Let $A$ be the first term,

$D$ be the common difference

and $B$ be the $(2n+1)th$ term of the $A.P.$

then $B=A+2nD\Rightarrow D=\frac{B-A}{2n}$

Also, $a=A+(n+1-1)=\frac{A+B}{2}$

Similarly, $b=\sqrt{AB}$ and $c=\frac{2AB}{A+B}$

Clearly, we have $b^2=ac$