If x_1,x_2,,x_n and 1h_1,1h_2,,1h_n are two A.P.'s such that x_3=h_2=8 and x_8=h_7=20, the…

Question

If $x_1,x_2,\ldots,x_n$ and $\dfrac{1}{h_1},\dfrac{1}{h_2},\ldots,\dfrac{1}{h_n}$ 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:

Accepted Answer

For $\{x_n\}$: $5d_1=x_8-x_3=12 \Rightarrow d_1=2.4$, so $x_5=x_3+2d_1=12.8$.

For $\{1/h_n\}$: $5d_2 = \dfrac{1}{20}-\dfrac18 = -\dfrac{3}{40} \Rightarrow d_2=-\dfrac{3}{200}$.

$$\frac{1}{h_{10}} = \frac{1}{h_7}+3d_2 = \frac{1}{20}-\frac{9}{200} = \frac{1}{200} \implies h_{10}=200.$$

$$x_5\cdot h_{10} = 12.8\times200 = 2560.$$

Solutions · 1