Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If <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> show that a, b, c are in A.P.

Answer
space space space tan open parentheses straight A over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis over denominator straight s left parenthesis straight s minus straight a right parenthesis end fraction end root comma space space space tan open parentheses straight B over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight a right parenthesis over denominator straight s left parenthesis straight s minus straight b right parenthesis end fraction end root comma space space tan open parentheses straight C over 2 close parentheses space equals space square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis over denominator straight s left parenthesis straight s minus straight c right parenthesis end fraction end root
tan open parentheses straight A over 2 close parentheses plus tan open parentheses straight C over 2 close parentheses space equals space 2 over 3 cot open parentheses straight B over 2 close parentheses
rightwards double arrowsquare root of fraction numerator left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis over denominator straight s left parenthesis straight s minus straight a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis straight s minus straight a right parenthesis left parenthesis straight s minus straight b right parenthesis over denominator straight s left parenthesis straight s minus straight c right parenthesis end fraction end root space equals space 2 over 3 square root of fraction numerator straight s left parenthesis straight s minus straight b right parenthesis over denominator left parenthesis straight s minus straight c right parenthesis left parenthesis straight s minus straight a right parenthesis end fraction end root
rightwards double arrow  square root of fraction numerator straight s minus straight b over denominator straight s end fraction end root open square brackets square root of fraction numerator straight s minus straight c over denominator straight s minus straight a end fraction end root plus square root of fraction numerator straight s minus straight a over denominator straight s minus straight c end fraction end root close square brackets space equals space 2 over 3 square root of fraction numerator straight s left parenthesis straight s minus straight b right parenthesis over denominator left parenthesis straight s minus straight c right parenthesis left parenthesis straight s minus straight a right parenthesis end fraction end root

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      2 straight s minus straight a minus straight c space equals space 2 over 3 straight s                     rightwards double arrow   left parenthesis straight a plus straight b plus straight c right parenthesis minus left parenthesis straight a plus straight c right parenthesis space equals space 1 third left parenthesis straight a plus straight b plus straight c right parenthesis

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   2b = a + c
rightwards double arrow      space space straight b equals fraction numerator straight a plus straight c over denominator 2 end fraction
Hence, a, b, c are in A.P.

More Chapters from Principle of Mathematical Induction