Permutations and Combinations

Permutations and Combinations

Question

Insert 5 G.M.'s between 32 over 9 space and space 81 over 2

Answer

Let <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> be five G.M.'s between 32 over 9 and 81 over 2
∴ 32 over 9 comma space straight G subscript 1 comma space straight G subscript 2 comma space straight G subscript 3 comma space straight G subscript 4 comma space straight G subscript 5 comma space 81 over 2 are in G.P.
Here, straight a space equals space 32 over 9 comma  Let r be the common ratio
Now, straight t subscript 7 space equals space 81 over 2 space rightwards double arrow space ar to the power of 7 minus 1 end exponent space equals space 81 over 2 space rightwards double arrow space 32 over 9 straight r to the power of 6 space equals space 81 over 2 space rightwards double arrow space straight r to the power of 6 space equals space 729 over 64 space rightwards double arrow space straight r to the power of 6 space equals space open parentheses 3 over 2 close parentheses to the power of 6 space rightwards double arrow space straight r space equals space plus-or-minus 3 over 2
When r = 3 over 2 comma  straight a space equals space 32 over 9 comma then
                             straight G subscript 1 space equals space straight t subscript 2 space equals space ar space equals space 32 over 9.3 over 2 space equals space 16 over 3
                              straight G subscript 2 space equals space straight t subscript 3 space equals space ar squared space equals space 32 over 9 open parentheses 3 over 2 close parentheses squared space equals space 32 over 9.9 over 4 space equals space 8
                              straight G subscript 3 space equals space straight t subscript 4 equals ar cubed space equals space 32 over 9 open parentheses 3 over 2 close parentheses cubed space equals space 32 over 9.27 over 8 equals 12
                             straight G subscript 4 space equals space straight t subscript 5 space equals ar to the power of 4 space equals space 32 over 9 open parentheses 3 over 2 close parentheses to the power of 4 space equals space 32 over 9.81 over 16 space equals space 18
                             <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>
When straight r equals fraction numerator negative 3 over denominator 2 end fraction comma space space space straight a space equals space 32 over 9 comma  then
                              straight G subscript 1 space equals space straight t subscript 2 space equals space ar space equals space 32 over 9 open parentheses fraction numerator negative 3 over denominator 2 end fraction close parentheses space equals space fraction numerator negative 16 over denominator 3 end fraction
                             straight G subscript 2 space equals space straight t subscript 3 space equals space ar squared space equals space 32 over 9 open parentheses 3 over 2 close parentheses squared space equals space 32 over 9. space 9 over 4 space equals space 8
                             straight G subscript 3 space equals space straight t subscript 4 space equals space ar cubed space equals space 32 over 9 open parentheses fraction numerator negative 3 over denominator 2 end fraction close parentheses cubed space equals space 32 over 9 open parentheses fraction numerator negative 27 over denominator 8 end fraction close parentheses space equals space minus 12
                             straight G subscript 4 space equals space straight t subscript 5 space equals space ar to the power of 4 space equals space 32 over 9 open parentheses fraction numerator negative 3 over denominator 2 end fraction close parentheses to the power of 4 space equals space 32 over 9 open parentheses 81 over 16 close parentheses space equals space 18
                            space space space straight G subscript 5 space equals space straight t subscript 6 space equals space ar to the power of 5 space equals space 32 over 9 open parentheses fraction numerator negative 3 over denominator 2 end fraction close parentheses to the power of 5 space equals space 32 over 9 open parentheses fraction numerator negative 243 over denominator 32 end fraction close parentheses space equals space minus 27
                     

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