Sets

Sets

Question

Prove the following by using the principle of mathematical induction for all straight n element of straight N.

fraction numerator 1 over denominator 3.5 end fraction space plus space fraction numerator 1 over denominator 5.7 end fraction space plus space fraction numerator 1 over denominator 7.9 space end fraction space plus space......... space plus space fraction numerator 1 over denominator left parenthesis 2 straight n plus 1 right parenthesis left parenthesis 2 straight n plus 3 right parenthesis end fraction space equals space fraction numerator straight n over denominator 3 space left parenthesis 2 straight n space plus space 3 right parenthesis end fraction

Answer

Let P(n) : fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus...... plus fraction numerator 1 over denominator left parenthesis 2 straight n plus 1 right parenthesis left parenthesis 2 straight n plus 3 right parenthesis end fraction space equals space fraction numerator straight n over denominator 3 space left parenthesis 2 straight n plus 3 right parenthesis end fraction
I.   For n = 1,
     space space straight P left parenthesis 1 right parenthesis space colon space fraction numerator 1 over denominator 3.5 end fraction space equals space fraction numerator 1 over denominator 3 left parenthesis 2.1 space plus 3 right parenthesis end fraction space rightwards double arrow space space fraction numerator 1 over denominator 3.5 end fraction space equals space fraction numerator 1 over denominator 3.5 end fraction space rightwards double arrow space 1 over 15 space equals space 1 over 15

∴     P(1) is true
II.  Let the statement be true for n = m,  straight m space element of space straight N

∴    P(m): fraction numerator 1 over denominator 3.5 end fraction space plus space fraction numerator 1 over denominator 5.7 end fraction plus space fraction numerator 1 over denominator 7.9 end fraction plus space.......... plus space fraction numerator 1 over denominator left parenthesis 2 straight m plus 1 right parenthesis left parenthesis 2 straight m plus 3 right parenthesis end fraction space equals space fraction numerator straight m over denominator 3 left parenthesis 2 straight m plus 3 right parenthesis end fraction                          ...(i)
III.  For n = m + 1,
       <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>
or  fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus......... plus fraction numerator 1 over denominator left parenthesis 2 straight m plus 3 right parenthesis space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m space plus 5 right parenthesis end fraction
or    fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis 2 straight m space plus space 3 right parenthesis end fraction space plus space fraction numerator 1 over denominator left parenthesis 2 straight m space plus space 3 right parenthesis space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction   ... (ii)

                                (Note that the last but one term in (ii) is always the same as last term in (i)]
        From (i), fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 3 right parenthesis end fraction space equals space fraction numerator straight m over denominator 3 space left parenthesis 2 straight m plus 3 right parenthesis end fraction

∴    straight P left parenthesis straight m plus 1 right parenthesis colon space fraction numerator straight m over denominator 3 space left parenthesis 2 straight m plus 3 right parenthesis end fraction plus space fraction numerator 1 over denominator left parenthesis 2 straight m plus 3 right parenthesis space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction
rightwards double arrow space space space space space space space space fraction numerator 1 over denominator left parenthesis 2 straight m plus 3 right parenthesis end fraction open square brackets straight m over 3 plus fraction numerator 1 over denominator 2 straight m plus 5 end fraction close square brackets space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction space rightwards double arrow space fraction numerator 1 over denominator 2 straight m plus 3 end fraction open square brackets fraction numerator 2 straight m squared plus 5 straight m plus 3 over denominator 3 space left parenthesis 2 straight m space plus 5 right parenthesis end fraction close square brackets space equals space fraction numerator straight m plus 1 over denominator 3 left parenthesis 2 straight m space plus 5 right parenthesis end fraction
space space rightwards double arrow space space space fraction numerator 1 over denominator 2 straight m plus 3 end fraction open square brackets fraction numerator left parenthesis 2 straight m plus 3 right parenthesis space left parenthesis straight m plus 1 right parenthesis over denominator 3 space left parenthesis 2 straight m space plus 5 right parenthesis end fraction close square brackets space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction rightwards double arrow space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction
        which is true

∴      P (m + 1) is true.

∴      P(m) is true rightwards double arrow P(m + 1) is true.
Hence, by the principal of mathematical induction, P(n) is true for all straight n element of space straight N.

More Chapters from Sets