Question

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

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

Answer 1

Answer: (B), (D)

The correct options are
B $a\ge b\ge c$
D $ac=b^2$

If a sequence has $2n+1$ terms in total, then the $n+1^{th}$ term is the middle term.

Let the 1st term is $a_1$ and the $2n+1^{th}$ term is $a_{2n+1}$

For A.P. $a_{n+1}=\frac{a_1+a_{2n+1}}{2}$

$\Rightarrow a=\frac{a_1+a_{2n+1}}{2}\dots \left(1\right)$

Similarly for G.P. $\sqrt{a_1.a_{2n+1}}=a_{n+1}$

$\Rightarrow b=\sqrt{a_1.a_{2n+1}}\dots \left(2\right)$

Finally for H.P. $a_{n+1}=\frac{2a_1.a_{2n+1}}{a_1+a_{2n+1}}$

$\Rightarrow c=\frac{2a_1.a_{2n+1}}{a_1+a_{2n+1}}\dots \left(3\right)$

From $\left(1\right),\left(2\right),\left(3\right)$ we get

$b^2=ac$

and $a\ge b\ge c$ ( $\because A.M.\ge G.M.\ge H.M.$)