Permutations and Combinations

Permutations and Combinations

Question

If 4th, 10th and 16th terms of a GP. are x, y and z respectively, Prove that y, z are in GP.

Answer

Let a be the first term and r be the common ratio of G.P.
Given straight t subscript 4 space equals space straight x comma space space space straight t subscript 10 space equals space straight y comma space space straight t subscript 16 space equals space straight z
rightwards double arrow                         ar cubed space equals space straight x                                       ...(i)
                             ar to the power of 9 space equals space straight y                                       ...(ii)
                           space space ar to the power of 15 space equals space straight z                                       ...(iii)
Multiply (i) and (iii), we get <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>
or                        straight a squared straight r to the power of 18 space equals space xz   or          xz space equals space left parenthesis ar to the power of 9 right parenthesis squared space equals space straight y squared            [By using (ii)]

∴                     space space xz space equals space straight y squared          or          straight y over straight x space equals space straight z over straight y  or  x, y, z are in G.P.

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