Sets

Sets

Question

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

n (n + 1) (n + 5) is a multiple of 3.

Answer

Let P(n): n (n + 1) (n + 5) is a multiple of 3.
I.         For n = 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 a multiple of 3
rightwards double arrow      1 (2) (6) is a multiple of 3 rightwards double arrow 12 is a multiple of 3.
           which is true.

∴          P(n) is true for n = 1.
II.       Suppose P (n) is true for n = m
rightwards double arrow     P(m) : m(m + 1) (m + 5) is a multiple of 3 rightwards double arrow m (m + 1) (m + 5) = 3k
rightwards double arrow     straight m cubed plus 6 straight m squared plus 5 straight m space equals space 3 straight k
rightwards double arrow      straight m cubed space equals space 3 straight k space minus space 6 straight m squared minus 5 straight m                                         ...(i)
III.      For  n = m + 1,
           P(m + 1): (m + 1) (m + 1 + 1) (m + 1 + 5) is a multiple of 3.
           (m + 1) (m + 2) (m + 6) is a multiple of 3.
           Now, (m + 1) (m + 2) (m + 6) = (m + 1)(straight m squared plus 8 straight m plus 12) =
                                                                    straight m cubed plus 9 straight m squared plus 20 straight m plus 12
           = 3 straight k minus 6 straight m squared minus 5 straight m space plus space 9 straight m squared space plus space 20 straight m space plus space 12                 [BY (i)]
           = 3 straight k space plus space 3 straight m squared space plus space 15 straight m space plus space 12 space equals space 3 left square bracket straight k plus straight m squared plus 5 straight m plus 4 right square bracket equals 3 straight k apostrophe
               where <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        (m + 1) (m + 2) (m + 6) is a multiple of 3.

∴           P(m + 1) is true.

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



More Chapters from Sets