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

Question

Let l_1,l_2, ... ,l_100 be consecutive terms of and arithmetic progression with common difference d_1, and let w_1,w_2, ... ,w_100 be consecutive terms of another arithmetic progression with common difference d_2, where d_1d_2 = 10. For each i = 1,2, ... ,100, let R_i be a rectangle with length l_i, width w_i and area A_i. If A_51 - A_50 = 1000, then the value of A_100 - A_90 is \_\_\_\_.

DivineJEE · Accepted Answer

A_51-A_50 = (l_1+50 d_1)(w_1+50 d_2) - (l_1+49 d_1)(w_1+49 d_2) = l_1 d_2 + w_1 d_1 + (50^2-49^2)d_1 d_2 \ = 1000 Therefore l_1 d_2 + w_1 d_1 = 10 qquad (Because d_1 d_2=10) Now, A_100-A_90 = (l_1+99 d_1)(w_1+99 d_2) - (l_1+89 d_1)(w_1+89 d_2) = 10(l_1 d_2 + w_1 d_1) + (99^2-89^2)d_1 d_2 = (10)(10) + (1880)(10) = 18900

Explanation

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