Permutations And Combinations

  • Question
    CBSEENMA11013204

    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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#6 {main}</pre>

    Solution


    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

                  <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
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#6 {main}</pre>
    rightwards double arrow               space space EF space equals space square root of 128 space equals space 8 square root of 2
    Area of first square = <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
    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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