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Permutations and Combinations

Permutations and Combinations

Question
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If straight a squared comma space straight b squared comma space straight c squared are in A.P., then prove that fraction numerator straight a over denominator straight b plus straight c end fraction comma space fraction numerator straight b over denominator straight c plus straight a end fraction comma space fraction numerator straight c over denominator straight a plus straight b end fraction are in A.P.

Answer
fraction numerator straight a over denominator straight b plus straight c end fraction comma space fraction numerator straight b over denominator straight c plus straight a end fraction comma space fraction numerator straight c over denominator straight a plus straight b end fraction will be in A.P.
if fraction numerator straight a over denominator straight b plus straight c end fraction plus 1 comma space space fraction numerator straight b over denominator straight c plus straight a end fraction plus 1 comma space space fraction numerator straight c over denominator straight a plus straight b end fraction plus 1 will be in A.P.                (By adding 1 to each number)
if  fraction numerator straight a plus straight b plus straight c over denominator straight b plus straight c end fraction comma space space fraction numerator straight a plus straight b plus straight c over denominator straight c plus straight a end fraction comma space fraction numerator straight a plus straight b plus straight c over denominator straight a plus straight b end fraction will be in A.P.
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> will be in A.P.                   (By dividing each number a + b + c)
if     space space space space fraction numerator 1 over denominator straight c plus straight a end fraction minus fraction numerator 1 over denominator straight b plus straight c end fraction space equals space fraction numerator 1 over denominator straight a plus straight b end fraction minus fraction numerator 1 over denominator straight c plus straight a end fraction  if    space space fraction numerator straight b plus straight c minus straight c minus straight a over denominator left parenthesis straight c plus straight a right parenthesis space left parenthesis straight b plus straight c right parenthesis end fraction space equals space fraction numerator straight c plus straight a minus straight a minus straight b over denominator left parenthesis straight a plus straight b right parenthesis space left parenthesis straight c plus straight a right parenthesis end fraction
if       fraction numerator straight b minus straight a over denominator straight c plus straight b end fraction space equals space fraction numerator straight c minus straight b over denominator straight b plus straight a end fraction                       if        (b - a) (b + a) = (c - b) (c + b)
if      space space space space straight b squared minus straight a squared space equals space straight c squared minus straight b squared
if        space space straight a squared comma space straight b squared comma space straight c squared   are in A.P. which is true.
Hence,space space space space space fraction numerator straight a over denominator straight b plus straight c end fraction comma space space fraction numerator straight b over denominator straight c plus straight a end fraction comma space space fraction numerator straight c over denominator straight a plus straight b end fraction are in A.P.

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