Question

Let $a_1,a_2,\dots ,a_{10}$ be an A.P. and $h_1,h_2,\dots ,h_{10}$ be in H.P. If $a_1=h_1=2$ and $a_{10}=h_{10}=3$, then the value of $a_4{h}_7$ is

Answer 1

$a_1,a_2,\dots ,a_{10}$ be an A.P.
Let the common difference be $d$,
$a_{10}=a_1+9d\Rightarrow 3=2+9d\Rightarrow d=\frac{1}{9}{a}_4=2+3d=2+\frac{1}{3}=\frac{7}{3}$

$h_1,h_2,\dots ,h_{10}$ be in H.P.
So, $\frac{1}{h_1},\frac{1}{h_2},\dots ,\frac{1}{h_{10}}$ be in A.P.
Let the common difference be $D$,
$\frac{1}{h_{10}}=\frac{1}{h_1}+9d\Rightarrow \frac{1}{3}=\frac{1}{2}+9d\Rightarrow d=-\frac{1}{54}\frac{1}{h_7}=\frac{1}{h_1}+6d\Rightarrow \frac{1}{h_7}=\frac{1}{2}-\frac{1}{9}\Rightarrow h_7=\frac{18}{7}$
Therefore,
$a_4{h}_7=\frac{7}{3}\times \frac{18}{7}=6$