l₁, l₂, l₃,.....l₁₀₀ be consecutive terms of an arithmetic progression with common difference d₁ and w₁,w₂,w₃ .....w₁₀₀be consecutive terms of another arithmetic progression with common difference d₂ where d₁d₂ = 10. For each i = 1,2,3.....100. R_i be a rectangle with length l_i. width w_i and area A_i . If A₅₁ - A₅₀ = 1000 then the value of $\frac{1}{100} (A_{100} - A_{90})$ is

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answer is 189.

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Detailed Solution

$A_{51} - A_{50} = (l_{1} + 50 d_{1}) (w_{1} + 50 d_{2}) - (l_{1} + 49 d_{1}) (w_{1} + 49 d_{2}) = 1000 \begin{array}{l} \Rightarrow l_{1} d_{2} + w_{1} d_{1} = 10 (d_{1} d_{2} = 10) \\ \begin{aligned} A_{100} - A_{90} & = (l_{1} + 99 d_{1}) (w_{1} + 99 d_{2}) - (l_{1} + 89 d_{1}) (w_{1} + 89 d_{2}) \end{aligned} \\ \begin{array}{l} = 10 (l_{1} d_{2} + w_{1} d_{1}) + (99^{2} - 89^{2}) d_{1} d_{2} \\ = 100 + 1880 (10) = 18900 \end{array} \frac{1}{100} (A_{100 -} A_{90}) = 189 \end{array}$