Question

If $x_1,x_2,.....x_n$ and $\frac{1}{h_1},\frac{1}{h_2},\dots ,\frac{1}{h_n}$ 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 :

Answer 1

Formula,

$x_n=a+(n-1)d$

Given,

$x_3=a+2d=8$....(1)

$x_8=a+7d=20$......(2)

solving (1) and (2), we get,

$d=\frac{12}{5},a=\frac{16}{5}$

$x_5=a+4d$

$=\frac{16}{5}+4\times \frac{12}{5}$

$=\frac{64}{5}$

$x_{10}=a+9d$

$=\frac{16}{5}+9\times \frac{12}{5}$

$=\frac{124}{5}$

$\Rightarrow h_{10}=\frac{5}{124}$

$x_5\cdot h_{10}=\frac{64}{5}\cdot \frac{5}{124}$

$=\frac{16}{31}$