Let $l_{1}, l_{2}, ...... l_{100}$ be consecutive terms of an A.P with common difference $d_{1}$ and let $w_{1}, w_{2}, .... w_{100}$ be the consecutive terms of another arthmatics progression with common difference $d_{2}$ where $d_{1} d_{2} = 10$ for each $i = 1, 2, ..., 100$ let $R_{1}$ be a rectangle with length $L_{i}$ , width $W_{i}$ and area $A_{i}$ . If $A_{51} - A_{50} = 1000$ value of $A_{100} - A_{90}$ is

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18,900

19,800

18,700

18,800

answer is B.

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Detailed Solution

For A.P $l_{1}, l_{2}, ...... l_{100}$
Let $T_{1} = a$ and common difference $= d_{1}$ and similarly for A.P $w_{1}, w_{2}, .... w_{100}$
$T_{1} = b$ and common difference $= d_{2}$
$A_{51} - A_{50} = l_{51} w_{51} - l_{50} w_{50} = (a + 50 d_{1}) (b + 50 d_{2}) - (a + 49 d_{1}) (b + 49 d_{2}) = 50 b d_{1} + 50 a d_{2} + 2500 d_{1} d_{2} - 49 a d_{2} - 49 b d_{1} - 2401 d_{1} d_{2} \Rightarrow ∴ b d_{1} + a d_{2} = 10...... (1) (A s d_{1} d_{2} = 10) ∴ A_{100} - A_{90} = l_{100} w_{100} - l_{90} w_{90} = (a + 99 d_{1}) (b + 99 d_{2}) - (a + 89 d_{1}) (b + 89 d_{2}) = 99 b d_{1} + 99 a d_{2} + 99^{2} d_{1} d_{2} - 89 b d_{1} - 89 a d_{2} - 89^{2} d_{1} d_{2} \begin{array}{l} \Rightarrow 10 (b d_{1} + a d_{2}) + 1800 d_{1} d_{2} \\ \Rightarrow 10 (10) + 1800 \Rightarrow 18, 900 \end{array}$