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

Answer 1

Answer: (B), (D)

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

Given first and $(2n-1{)}^{th}$ terms of $AP,GP$ and $HP$ are equal.

And $n^{th}$ terms are $a,b$ and $c$ respectively.

Let first and $(2n-1{)}^{th}$ terms are $x$ and $y$ respectively.

$\Rightarrow a=\frac{x+y}{2}$

$\Rightarrow b=\sqrt{xy}$

$\Rightarrow c=\frac{2xy}{x+y}$

$\therefore a\ge b\ge c$ and $b^2=ac$ Since, $AM\ge GM\ge HM$ and $G{M}^2=\left(AM\right)\left(HM\right)$

Hence, options B and D.