Permutations and Combinations

Permutations and Combinations

Question

Find the sum to n terms of the sequence space space open curly brackets straight t subscript straight n close curly brackets given by <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>

Answer

The given sequence is space space open curly brackets straight t subscript straight n close curly brackets  defined byspace space space space space space straight t subscript straight n equals space 2 to the power of straight n space plus space 3 straight n
                           straight t subscript 1 space equals space 2 to the power of 1 space plus space 3. space 1 space equals space 2 to the power of 1 space plus space 3
                           straight t subscript 2 space equals space 2 squared space plus space 3. space 2 space equals space 2 squared plus 6
                            straight t subscript 3 space equals space 2 cubed space plus space 3. space 3 space equals space 2 cubed space plus space 9
Let space space straight S subscript straight n  denote the sum of n terms

∴                             <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>
                                   = left parenthesis 2 to the power of 1 plus 3 right parenthesis space plus space left parenthesis 2 squared plus 6 right parenthesis space plus space left parenthesis 2 cubed plus 9 right parenthesis space plus space........... space plus space left parenthesis 2 to the power of straight n plus 3 straight n right parenthesis
                                   equals left parenthesis 2 to the power of 1 plus 2 squared plus 2 cubed plus...... plus 2 to the power of straight n right parenthesis space plus space left parenthesis 3 plus 6 plus 9 plus....... plus 3 straight n right parenthesis
Since the series in the first bracket is a G.P. series with 1st term 2, common ratio 2 and the series in the second bracket is an A.P. with 1st term 3, common difference 3.

∴             straight S subscript straight n space equals space fraction numerator 2 left parenthesis 2 to the power of straight n minus 1 right parenthesis over denominator 2 minus 1 end fraction space plus space straight n over 2 left square bracket 3 plus 3 straight n right square bracket space equals space 2 to the power of straight n plus 1 end exponent minus 2 space plus space fraction numerator 3 straight n over denominator 2 end fraction left parenthesis straight n plus 1 right parenthesis


                                   
                                  


                          
      

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