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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 :

Option 1)
$2560$

Option 2)
$2650$

Option 3)
$3200$

Option 4)
$1600$

Answers (2)

best_answer

As we learned

General term of an A.P. -

$T_{n}= a+\left ( n-1 \right )d$

- wherein

$a\rightarrow$ First term

$n\rightarrow$ number of term

$d\rightarrow$ common difference

AP₁ : x₁, x₂, x₃ --- difference = d₁

AP₂ : $\frac{1}{h_{1}},\frac{1}{h_{2}},\frac{1}{h_{3}} \rightarrow difference = d_{2}$

$x_{8}-x_{3}=20-8=5d_{1}\Rightarrow d_{1}=\frac{12}{5}$

$\frac{1}{h_{7}}-\frac{1}{h_{2}}=5d_{2}=\frac{1}{20}-\frac{1}{8}$

$5d_{2}=\frac{2-5}{40}\Rightarrow d_{2}=-\frac{3}{200}$

Also $x_{3}=8=x_{5}-\frac{24}{5}\Rightarrow x_{5}=\frac{64}{5}$

Also $\frac{1}{h_{10}}=\frac{1}{h_{7}}+3d=\frac{1}{20}-\frac{9}{200}$

$h_{10}=200$

Thus $x_{5}h_{10}=\frac{64}{5}\times 200=2560$

Option 1)

$2560$

Option 2)

$2650$

Option 3)

$3200$

Option 4)

$1600$

Posted by

Himanshu

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