Permutations And Combinations

  • Question
    CBSEENMA11012983

    The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r) a + (r – p) b + (p – q)c = 0

    Solution

    Let A be the first term and D be the c.d. of an A.P.
                    straight t subscript straight p space equals straight a space rightwards double arrow space straight A space plus space left parenthesis straight p minus 1 right parenthesis straight D space equals space straight a              ...(i)
    space space straight t subscript straight q space equals space straight b rightwards double arrow space straight A space plus space left parenthesis straight q minus 1 right parenthesis straight D space equals space straight b                    ...(ii)
    <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>                        ...(iii)
    L.H.S. = (q - r) a + (r - p)b + (p - q)c = (q - r) [A + (p - 1)D]
                                        + (r - p) [A + (q - 1)D + (p - q) [A + (r - 1)D]
                              (By using values of a, b, c from (i), (ii), (iii) respectively]
             = A(q - r) + (q - r) (p - 1)D + A (r - p) + (r - p) (q - 1)D + A(p - q) + (p - q) (r - 1)D
             = A(q - r + r - p + p - q) + D[Pq - q - rp + r + rq - r - pq + p + pr - p - qr + q]
             = A(0) + D(0) = 0 = R.H.S.

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