Permutations and Combinations

Find the sum to n terms of the sequence given by - Wired Faculty

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 $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Solution

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

∴ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
 = $left parenthesis 2 to the power of 1 plus 3 right parenthesis space plus space left parenthesis 2 squared plus 6 right parenthesis space plus space left parenthesis 2 cubed plus 9 right parenthesis space plus space........... space plus space left parenthesis 2 to the power of straight n plus 3 straight n right parenthesis$
 $equals left parenthesis 2 to the power of 1 plus 2 squared plus 2 cubed plus...... plus 2 to the power of straight n right parenthesis space plus space left parenthesis 3 plus 6 plus 9 plus....... plus 3 straight n right parenthesis$
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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