Sets

Sets

Question

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

10 to the power of 2 straight n minus 1 end exponent space plus space 1 is divisible by 11.

Answer

Let P(n) : 10 to the power of 2 straight n minus 1 end exponent space plus space 1 is divisible by 11
I.             For n = 1,
            P(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> is divisible by 11
rightwards double arrow        101 + 1 is divisible by 11 rightwards double arrow 11 is divisible by 11

∴           P(1) is true
II.          Suppose the statement is true for n = m, <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>

∴         P(m) : 10 to the power of 2 straight m minus 1 end exponent space plus space 1 is divisible by 11.
rightwards double arrow       10 to the power of 2 straight m minus 1 end exponent space plus space 1 space equals space 11 straight k comma space straight k space element of space straight Z
rightwards double arrow       10 to the power of 2 straight m minus 1 end exponent space equals space 11 straight k space minus space 1                                         ...(i)
III.         For n = m + 1,
            straight P left parenthesis straight m space plus space 1 right parenthesis space colon space space 10 to the power of 2 left parenthesis straight m plus 1 right parenthesis minus 1 end exponent plus 1 is divisible by 11.       ...(ii)
            Now, 10 to the power of 2 left parenthesis straight m plus 1 right parenthesis minus 1 end exponent plus 1 space equals space 10 to the power of 2 straight m minus 1 end exponent. space 10 squared space plus space 1 space equals space left parenthesis 11 straight k space minus space 1 right parenthesis. space 10 squared space plus space 1       [By (i)]
            =1100 straight k space minus space 100 space plus space 1 space equals space 1100 straight k space minus space 99 space equals space 11 space left parenthesis 100 straight k space minus space 9 right parenthesis space equals space 11 straight k apostrophe
                                                                          where k' = 100k - 9 space element of space straight Z

∴           10 to the power of 2 left parenthesis straight m plus 1 right parenthesis minus 1 end exponent space plus space 1 is divisible by 11


rightwards double arrow           P (m + 1) is true.

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


           



          

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