Question

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

Answer: (A)

$2560$

Let $d_1$ be the common difference of the A.P. $x_1,x_2,...$, then $x_8-x_3=5d_1=12\Rightarrow d_1=\frac{12}{5}=2.4$

$\Rightarrow x_5=x_3+2d_1=8+2\times \frac{12}{5}=12.8$

Let $d_2$ be the common difference of the other sequence then $5d_2=\frac{1}{20}-\frac{1}{8}=\frac{-3}{40}\Rightarrow d_2=\frac{-3}{200}$

$\Rightarrow \frac{1}{h_{10}}=\frac{1}{h_7}+3d_2=\frac{1}{200}\Rightarrow h_{10}=200$

$\Rightarrow x_5\cdot h_{10}=12.8\times 200=2560$

So option A is the correct answer.