Permutations and Combinations

Permutations and Combinations

Question

Find the sum of n terms of the series whose nth term is

space space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus...... plus straight n cubed over denominator straight n end fraction.

Answer

Here, straight T subscript straight n space equals space fraction numerator 1 cubed plus 2 cubed plus.... plus straight n cubed over denominator straight n end fraction space equals space fraction numerator sum straight n cubed over denominator straight n end fraction space equals space fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 straight n end fraction space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction space equals space straight n over 4 left parenthesis straight n squared plus 2 straight n plus 1 right parenthesis
rightwards double arrow    <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>
∴     straight S subscript straight n space equals space sum from straight k equals 1 to straight n of straight T subscript straight k space equals space sum from straight k equals 1 to straight n of open parentheses 1 fourth straight k cubed plus 1 half straight k squared plus 1 fourth straight k close parentheses space equals space 1 fourth sum from straight k equals 1 to straight n of straight k cubed plus 1 half sum from straight k equals 1 to straight n of straight k squared space plus space 1 fourth sum from straight k equals 1 to straight n of straight k
            equals 1 fourth fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction plus 1 half fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus 1 fourth fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction plus fraction numerator 2 straight n plus 1 over denominator 3 end fraction plus 1 half close square brackets
             equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction open square brackets fraction numerator 3 straight n squared plus 3 straight n plus 8 straight n plus 4 plus 6 over denominator 12 end fraction close square brackets space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 48 end fraction space left square bracket 3 straight n squared plus 11 straight n plus 10 right square bracket
           space space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 48 end fraction space left square bracket 3 straight n squared plus 6 straight n plus 5 straight n plus 10 space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 48 end fraction space left square bracket 3 straight n left parenthesis straight n plus 2 right parenthesis space plus space 5 left parenthesis straight n plus 2 right parenthesis right square bracket
             <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>
      

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