If $x_1, x_2, \ldots ., x_n$ and $\frac{1}{h_1}, \frac{1}{h^2}, \ldots . . \frac{1}{h_n}$ are two A.P's such…
If $x_1, x_2, \ldots ., x_n$ and $\frac{1}{h_1}, \frac{1}{h^2}, \ldots . . \frac{1}{h_n}$ are two A.P's such that $x_3=h_2=8$ and $x_8=h_7=20$, then $x_5 . h_{10}$ equals.
2560
2650
3200
1600
Solution
Suppose $d_1$ is the common difference of the A.P.
$x_1, x_2, \ldots . x_n$ then
$
\begin{aligned}
&\because x_8-x_3=5 d_1=12 \Rightarrow d_1=\frac{12}{5}=2.4 \\
&\Rightarrow x_5=x_3+2 d_1=8+2 \times \frac{12}{5}=12.8
\end{aligned}
$
Suppose $d_2$ is the common difference of the
A.P $\frac{1}{h_1}, \frac{1}{h_2}, \ldots . \cdot \frac{1}{h_n}$ then
$
\begin{aligned}
&5 d_2=\frac{1}{20}-\frac{1}{8}=\frac{-3}{40} \Rightarrow d_2=\frac{-3}{200} \\
&\because \frac{1}{h_{10}}=\frac{1}{h_7}+3 d_2=\frac{1}{200} \Rightarrow h_{10}=200 \\
&\Rightarrow x_5 . h_{10}=12.8 \times 200=2560
\end{aligned}
$