Permutations And Combinations

Question
CBSEENMA11013085

Sum the series

space space 2 over 9 space plus space 1 third plus space 1 half plus space............ space plus space 81 over 32

Solution

The given series is : <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()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#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>
Here,  space space straight a equals 2 over 9 comma space space straight r space equals space fraction numerator begin display style 1 third end style over denominator begin display style 2 over 9 end style end fraction space equals space 1 third.9 over 2 equals 3 over 2.
Let    l space equals space 81 over 32 be the nth term
Required sum  = fraction numerator straight a minus lr over denominator 1 minus straight r end fraction space equals space fraction numerator begin display style 2 over 9 end style minus begin display style 81 over 32 end style open parentheses begin display style 3 over 2 end style close parentheses over denominator 1 minus begin display style 3 over 2 end style end fraction space equals space fraction numerator begin display style 2 over 9 end style minus begin display style 243 over 64 end style over denominator begin display style fraction numerator negative 1 over denominator 2 end fraction end style end fraction space equals space fraction numerator 128 minus 2187 over denominator 9.64 end fraction open parentheses fraction numerator negative 2 over denominator 1 end fraction close parentheses space equals space fraction numerator 2059 over denominator 9. space 32 space end fraction space equals space 2059 over 288

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