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Maths-

General

Easy

Question

Concentric circles of radii 1, 2, 3,…,100 cm are drawn. The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. Then, the total area of the green regions in sq. cm is equal to

$1000 π$
$5050 π$
$4950 π$
$5151 π$

The correct answer is: $5050 pi$

$pi open square brackets open parentheses r subscript 2 end subscript superscript 2 end superscript minus r subscript 1 end subscript superscript 2 end superscript close parentheses plus open parentheses r subscript 4 end subscript superscript 2 end superscript minus r subscript 3 end subscript superscript 2 end superscript close parentheses plus horizontal ellipsis plus open parentheses r subscript 100 end subscript superscript 2 end superscript minus r subscript 99 end subscript superscript 2 end superscript close parentheses close square brackets$
$equals pi open square brackets r subscript 1 end subscript plus r subscript 2 end subscript plus r subscript 3 end subscript plus r subscript 4 end subscript plus horizontal ellipsis plus r subscript 100 end subscript close square brackets blank open parentheses because r subscript 2 end subscript minus r subscript 1 end subscript equals r subscript 4 end subscript minus r subscript 3 end subscript equals horizontal ellipsis equals r subscript 100 end subscript minus r subscript 99 end subscript equals 1 close parentheses$
$equals pi open square brackets 1 plus 2 plus 3 plus horizontal ellipsis plus r subscript 100 end subscript close square brackets$
$equals 5050 pi blank s q. c m$

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