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Mathematics · Sequence and Series

JEE Advanced 2022 — Paper 1 — Question 6

Let l1,l2,….,l100l_{1}, l_{2}, \ldots ., l_{100} be consecutive terms of an arithmetic with common difference d1d_{1}, and let

w1,w2,…..,w100w_{1}, w_{2}, \ldots . ., w_{100} be consecutive terms of another arithmetic progression with common difference d2d_{2},

where d1d2=10d_{1} d_{2}=10. For each i=1,2,….,100i=1,2, \ldots ., 100, let RiR_{i} be a rectangle with length lil_{i}, width wiw_{i}

and area AiA_{i}. If A51−A50=1000A_{51}-A_{50}=1000, then the value of A100−A90A_{100}-A_{90} is \qquad .

Answer: 18900

Numerical answer — enter this value.

Step-by-step solution

A51−A50=1000\mathrm{A}_{51}-\mathrm{A}_{50}=1000

l51 W51−l50 W50=100l_{51} \mathrm{~W}_{51}-l_{50} \mathrm{~W}_{50}=100

(l1+50 d1)(w1+50 d2)−(l1+49 d1)(w1+49 d2)=1000\left(l_{1}+50 \mathrm{~d}_{1}\right)\left(\mathrm{w}_{1}+50 \mathrm{~d}_{2}\right)-\left(l_{1}+49 \mathrm{~d}_{1}\right)\left(\mathrm{w}_{1}+49 \mathrm{~d}_{2}\right)=1000

l1 W1+50l1 d2+50 d1 W1+2500 d1 d2−l1 W1−49l1 d2−49 d1 W1−240 d1 d2=1000l_{1} \mathrm{~W}_{1}+50 l_{1} \mathrm{~d}_{2}+50 \mathrm{~d}_{1} \mathrm{~W}_{1}+2500 \mathrm{~d}_{1} \mathrm{~d}_{2}-l_{1} \mathrm{~W}_{1}-49 l_{1} \mathrm{~d}_{2}-49 \mathrm{~d}_{1} \mathrm{~W}_{1}-240 \mathrm{~d}_{1} \mathrm{~d}_{2}=1000

⇒l1 d2+d1 W1+99 d1 d2=1000\Rightarrow l_{1} \mathrm{~d}_{2}+\mathrm{d}_{1} \mathrm{~W}_{1}+99 \mathrm{~d}_{1} \mathrm{~d}_{2}=1000

⇒l1 d2+d1 W1=10\Rightarrow l_{1} \mathrm{~d}_{2}+\mathrm{d}_{1} \mathrm{~W}_{1}=10

A100−A90=(l100 W100)−(l90 W90)\mathrm{A}_{100}-\mathrm{A}_{90}=\left(l_{100} \mathrm{~W}_{100}\right)-\left(l_{90} \mathrm{~W}_{90}\right)

=(l1+99 d1)(w1+99 d2)−(l1+89 d1)(w1+89 d2)=\left(l_{1}+99 \mathrm{~d}_{1}\right)\left(\mathrm{w}_{1}+99 \mathrm{~d}_{2}\right)-\left(l_{1}+89 \mathrm{~d}_{1}\right)\left(\mathrm{w}_{1}+89 \mathrm{~d}_{2}\right)

=10 d1 W1+10l1 d2+1880 d1 d2=10 \mathrm{~d}_{1} \mathrm{~W}_{1}+10 l_{1} \mathrm{~d}_{2}+1880 \mathrm{~d}_{1} \mathrm{~d}_{2}

=10×10+18800=18900=10 \times 10+18800=18900

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2022
Paper
Paper 1
Subject
Mathematics
Chapter
Sequence and Series
Topic
Arithmetic Progression

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