Question

If $x_1,x_2,\dots ,x_n$ and $\frac{1}{h_1},\frac{1}{h_2},\dots ,\frac{1}{h_n}$ are two $A.P.P.s$ 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 $2560$

Let $m$ be the common difference of the AP

$\begin{array}{c}{x}_1,x_2\dots x_n\text{then}{x}_8-x_5=5m=12\\ \Rightarrow m=\frac{12}{5}=2.4\\ \Rightarrow x_5=x_3+2m=8+2\times \frac{12}{5}=12.8\end{array}$

Let $x$ be the common difference of the $AP$

$\begin{array}{cc}\Rightarrow n& =\frac{-3}{200}\\ \Rightarrow \frac{\hslash }{h_{10}}=\frac{1}{h_7}+3n=\frac{1}{200}\Rightarrow h_{10}=200& \\ \Rightarrow x_5\cdot h_{10}=12.8\times 200=2560& \end{array}$

Hence, $\left(A\right)$ is the correct option.