Permutations and Combinations

Permutations and Combinations

Question

If S1, S2, S3 ........Sp are the sums of n terms of p A.P.’s whose first terms are 1, 2,3, .....p and common differences are 1, 3, 5, .....(2p – 1) respectively. Show that

          space space straight S subscript 1 space plus space straight S subscript 2 space plus space........... space plus space straight S subscript straight p space equals space pn over 2 left parenthesis pn space plus space 1 right parenthesis

Answer

Here,                 space space straight S subscript 1= Sum of first n terms  of an A.P. with first term 1 and c.d. 1.
                             = straight n over 2 left square bracket 2. space 1 space plus space left parenthesis straight n space minus space 1 right parenthesis 1 right square bracket space equals space straight n over 2 left square bracket straight n plus 1 right square bracket
                         straight S subscript 2 equals Sum of first n terms of an A.P. with first term 2 and c.d. 3.
                             = straight n over 2 left square bracket 2.2 space plus space left parenthesis straight n space minus space 1 right parenthesis 3 right square bracket space equals space straight n over 2 left square bracket 3 straight n space plus space 1 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> =  Sum of first n terms of an A.P. with first term 3 and c.d. 5.
                             =straight n over 2 left square bracket 2.3 space plus space left parenthesis straight n minus 1 right parenthesis 5 right square bracket space equals straight n over 2 left square bracket 5 straight n space plus space 1 right square bracket
                            .............................................
                           ..............................................
                      straight S subscript straight p equalsSum of first n terms of an A.P. with first term p and c.d. (2p - 1)
                          <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>
∴   <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 n over 2 left parenthesis straight n plus 1 right parenthesis space plus space straight n over 2 left parenthesis 3 straight n plus 1 right parenthesis plus straight n over 2 left parenthesis 5 straight n plus 1 right parenthesis plus.......... plus straight n over 2 left square bracket left parenthesis 2 straight p minus 1 right parenthesis straight n plus 1 right square bracket
                                              = space space straight n over 2 left square bracket left parenthesis straight n plus 1 right parenthesis space plus space left parenthesis 3 straight n plus 1 right parenthesis space plus space left parenthesis 5 straight n plus 1 right parenthesis space plus....... plus left curly bracket left parenthesis 2 straight p space minus 1 right parenthesis straight n space plus 1 right curly bracket right square bracket   
                                              equals straight n over 2 left square bracket left parenthesis straight n space plus space 3 straight n space plus space 5 straight n plus...... plus space top enclose 2 straight p minus 1 end enclose straight n right parenthesis plus straight p right square bracket     
                                               = straight n over 2 left square bracket straight n left parenthesis 1 plus 3 plus 5 plus...... plus top enclose 2 straight p minus 1 end enclose right parenthesis space plus space straight p right square bracket
                                               = space space straight n over 2 open square brackets straight n. straight p over 2 left parenthesis 1 plus 2 straight p minus 1 right parenthesis plus straight p close square brackets space equals space straight n over 2 open square brackets np over 2.2 straight p plus straight p close square brackets        
                                                equals straight n over 2 left square bracket np squared plus straight p right square bracket space equals space fraction numerator np left parenthesis np plus 1 right parenthesis over denominator 2 end fraction
 

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