Permutations and Combinations

Permutations and Combinations

Question

If (m + l)th term of an A.P. is twice the (n + l)th term. Prove that (3m + l)th term is twice the (m + n + l)th term.

Answer

Let a be the first and d be the common difference of an A.P. Since (m + 1)th term of an A.P. is twice the (n + 1)th term.

∴          <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)
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#6 {main}</pre>
or    a + (m + 1 - 1)d = 2[a + (n + 1 - 1)d]
or          a + md = 2a + 2nd
or                       a = (m - 2n)d
Now,                 straight t subscript 3 straight m plus 1 end subscript space equals space straight a space plus space left parenthesis 3 straight m space plus space 1 minus 1 right parenthesis straight d space equals space straight a plus space 3 md space equals space left parenthesis straight m minus 2 straight n right parenthesis straight d space plus space 3 md space equals space left parenthesis straight m minus 2 straight n plus 3 straight m right parenthesis straight d
                                 equals left parenthesis 4 straight m minus 2 straight n right parenthesis straight d space equals space 2 left parenthesis 2 straight m minus straight n right parenthesis straight d
rightwards double arrow                  straight t subscript 3 straight m plus 1 end subscript space equals space 2 left parenthesis 2 straight m minus straight n right parenthesis straight d                                                     ...(i)
Now,            straight t subscript straight m plus straight n plus 1 end subscript space equals space straight a plus left parenthesis straight m plus straight n plus 1 minus 1 right parenthesis straight d space equals space left parenthesis straight m minus 2 straight n right parenthesis straight d space plus space left parenthesis straight m plus straight n right parenthesis straight d
                              equals space left parenthesis straight m minus 2 straight n space plus space straight m plus space straight n right parenthesis straight d
or                space space space space space straight t subscript straight m plus straight n plus 1 end subscript space equals space left parenthesis 2 straight m space minus space straight n right parenthesis straight d                                                    ...(ii)


From (i) and (ii), we have
                     <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>

∴   (3m + 1)th term is twice the (m + n + 1)th term of an A.P.

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