JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Sequences and Series

Let

l_{1}, l_{2}, \dots, l_{100}

be consecutive terms of an arithmetic progression with common difference

d_{1}

, and let

w_{1}, w_{2}, \dots, 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, \dots, 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 ____.

Answer: 18900

Solution

With

A_{i} = l_{i} w_{i}

l_{i} = l_{1} + (i - 1) d_{1}

and

w_{i} = w_{1} + (i - 1) d_{2}

. Expanding

A_{51} - A_{50} = 1000

gives

l_{1} d_{2} + w_{1} d_{1} + 99 d_{1} d_{2} = 1000

, and since

d_{1} d_{2} = 10

l_{1} d_{2} + w_{1} d_{1} = 1000 - 990 = 10

. Then

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}) + (9 9^{2} - 8 9^{2}) d_{1} d_{2} = 10 \times 10 + 1880 \times 10 = 18900

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Numerical answer