Permutations and Combinations

Permutations and Combinations

Question

If two G.M.'s <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)
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#6 {main}</pre> and one A.M. 'A' be inserted between two numbers, show that

2 straight A space equals space fraction numerator straight g subscript 1 superscript 2 over denominator straight g subscript 2 end fraction space plus space fraction numerator straight g subscript 2 superscript 2 over denominator straight g subscript 1 end fraction

Answer

Let a and b be two numbes, since g1 and g2 are two G.M.'s between a and b.

∴  a, straight g subscript 1 comma space space straight g subscript 2, b are in G.P.
Let r be common ratio

∴                    straight g subscript 1 space equals space ar comma space space straight g subscript 2 space equals space ar squared comma space straight b space equals space ar cubed
Since A is the A.M. between a and b

∴                      straight A space equals space fraction numerator straight a plus straight b over denominator 2 end fraction
rightwards double arrow                   2A = a + b                                                                ...(i)
                   R.H.S. = fraction numerator straight g subscript 1 superscript 2 over denominator straight g subscript 2 end fraction plus fraction numerator straight g subscript 2 superscript 2 over denominator straight g subscript 1 end fraction space equals space fraction numerator straight a squared straight r squared over denominator ar squared end fraction plus fraction numerator straight a squared straight r to the power of 4 over denominator ar end fraction space equals space straight a plus ar cubed space equals space straight a plus straight b space equals space 2 straight A  {By using (i)}
                             = L.H.S.

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