Question

If $x_1,x_2,\cdots ,x_n$ and $\frac{1}{h_1},\frac{1}{h_2},\cdots ,\frac{1}{h_n}$ are two $A.P.$s such that $x_3=h_2=8$ and $x_8=h_7=20,$ then $x_5\cdot h_{10}$ equals

• A. $2560$
• B. $2650$
• C. $3200$
• D. $1600$

Answer 1

The correct option is A $2560$

We know that,
$x_8=20\Rightarrow x_1+7d=20x_3=8\Rightarrow x_1+2d=8\Rightarrow 5d=12\Rightarrow x_5=x_1+4d=x_1+2d+2d\Rightarrow x_5=8+\frac{24}{5}=\frac{64}{5}$
Now,
$h_7=20\Rightarrow \frac{1}{h_1}+6d=\frac{1}{20}{h}_2=8\Rightarrow \frac{1}{h_1}+d=\frac{1}{8}\Rightarrow 5d=\frac{1}{20}-\frac{1}{8}\Rightarrow d=-\frac{3}{200}\Rightarrow \frac{1}{h_{10}}=\frac{1}{h_1}+9d=\frac{1}{h_1}+6d+3d\Rightarrow \frac{1}{h_{10}}=\frac{1}{20}-\frac{9}{200}\Rightarrow h_{10}=200$
Therefore,
$x_5{h}_{10}=\frac{64}{5}\times 200=2560$