Permutations And Combinations

  • Question
    CBSEENMA11013060

    Find all the sequences which are simultaneously in A.P. and GP.

    Solution

    Let the sequence space space open curly brackets straight t subscript straight n close curly brackets  be in A.P. as well as in G.P.
    Let straight t subscript straight n comma space space straight t subscript straight n plus 1 end subscript comma space straight t subscript straight n plus 2 end subscript be three consecutive terms of an A.P.

    ∴            straight t subscript straight n plus 1 end subscript space minus straight t subscript straight n space equals space straight t subscript straight n plus 2 end subscript minus straight t subscript straight n plus 1 end subscript space horizontal strike logical or space straight n greater or equal than 1
    or            space space 2 space straight t subscript straight n plus 1 end subscript space equals space straight t subscript straight n space end subscript plus space straight t subscript straight n plus 2 end subscript space horizontal strike logical or space straight n greater or equal than 1                        ...(i)
    Let r be the common ratio of G.P.

    ∴      straight t subscript straight n plus 1 end subscript over straight t subscript straight n space equals space fraction numerator straight t subscript straight n plus 2 end subscript over denominator straight t subscript straight n plus 1 end fraction space equals space straight r
    rightwards double arrow                  straight t subscript straight n plus 1 end subscript space equals space straight t subscript straight n straight r  and space space straight t subscript straight n plus 2 end subscript space equals space straight t subscript straight n plus 1 straight r space equals space straight t subscript straight n straight r squared

    Using the values of straight t subscript straight n plus 1 end subscript and space space straight t subscript straight n plus 2 end subscript in (i), we get
                       2 straight t subscript straight n straight r space equals space straight t subscript straight n space plus space straight t subscript straight n straight r squared              rightwards double arrow space space space space space space space space space space space space space space space space space space space space space 2 straight r space equals space 1 plus straight r squared
    rightwards double arrow        <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>                         rightwards double arrow            left parenthesis straight r minus 1 right parenthesis squared space equals space 0
    rightwards double arrow                  r = 1

    ∴                     space space straight t subscript 1 space equals space straight t subscript 2 space equals space straight t subscript 3 equals space.............          
    Hence, all the constant sequences are the only sequences which are in A.P. as well as in G.P.

    Delhi University

    NCERT Book Store

    NCERT Sample Papers

    Entrance Exams Preparation