Question

If $a_1,a_2;g_1,g_2$ and $h_1,h_2$ are two arithmetic, geometric and harmonic means respectively between two quantities $a$ and $b$, then $ab$ is equal to

• A. $g_1{g}_2$
• B. $h_1{h}_2$
• C. $a_1{h}_2$
• D. $a_2{h}_1$

Answer 1

Answer: (A), (C), (D)

$a_2{h}_1$

$a,a_1,a_2,b$ are in A.P.
Let common difference is $d$
Then $d=\frac{b-a}{3}$
$\Rightarrow a_1=\frac{2a+b}{3}$ and $a_2=\frac{a+2b}{3}$

$a,g_1,g_2,b$ are in G.P.
Let common ratio is $r$
Then $r={\left(\frac{b}{a}\right)}^{1/3}$
$\Rightarrow g_1=a^{2/3}{b}^{1/3}$ and $g_2=a^{1/3}{b}^{2/3}$

$a,h_1,h_2,b$ are in H.P.
$\Rightarrow \frac{1}{a},\frac{1}{h_1},\frac{1}{h_2},\frac{1}{b}$ are in A.P.
Let common difference is $D$
Then $D=\frac{a-b}{3ab}$
$\Rightarrow {\frac{1}{h}}_1=\frac{1}{a}+\left(\frac{a-b}{3ab}\right)=\frac{a+2b}{3ab}{h}_1=\frac{3ab}{a+2b}$
and $\frac{1}{h_2}=\frac{2a+b}{3ab}$
$\Rightarrow h_2=\frac{3ab}{2a+b}$

$\therefore a_1{h}_2=ab=a_2{h}_1=g_1{g}_2$