Permutations and Combinations

Permutations and Combinations

Question

A square is drawn by joining the mid-points of the sides of a given square. A third square is drawn inside the second square in the same way, and the process continues indefinitely. If a side of the first square is 16 cm., determine the sum of the areas of all the squares.

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Answer

Let ABCD be the given square of side 16cm as shown in the figure. Let EFGH be the second square drawn by joining the mid-points of the sides of given square ABCD.
         BF = EB = 8

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rightwards double arrow               space space EF space equals space square root of 128 space equals space 8 square root of 2
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Area of 2nd square = space space left parenthesis 8 square root of 2 right parenthesis squared space equals space 128 space sq. space cm.
Now, side of the third square = square root of left parenthesis 4 square root of 2 right parenthesis squared plus left parenthesis 4 square root of 2 right parenthesis squared space end root space equals space square root of 32 plus 32 end root space equals space 8
∴  Area of third square = left parenthesis 8 right parenthesis squared space equals space 64 sq. cm. Since the number of squares is infinite
∴  Sum of areas of all such squares = 256 + 128 + 64 + .......
                        equals fraction numerator 256 over denominator 1 minus begin display style 1 half end style end fraction                             open curly brackets because space straight a space equals space 256 comma space space straight r space equals space 128 over 256 space equals space 1 half close curly brackets
                   space space space space space space equals fraction numerator 256 over denominator begin display style 1 half end style end fraction space equals space 512

      

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