Sets

Sets

Question

Prove by mathematical induction that sum of cubes of three consecutive natural numbers is divisible by 9.

Answer

Let n, n+1, n+2 be three consecutive natural numbers.
Let P(n): straight n cubed space plus space left parenthesis straight n plus 1 right parenthesis cubed space plus space left parenthesis straight n plus 2 right parenthesis cubed is divisible by 9.
I.       For n = 1,
       straight P left parenthesis 1 right parenthesis space colon space 1 cubed space plus space 2 cubed space plus space 3 cubed is divisible by 9
rightwards double arrow  1 + 8 + 27 is divisible by 9 rightwards double arrow36 is divisible by 9
      which is true

∴     the statement is true for n = 1.
II.     Suppose the statement is true for n = m,  straight m space element of space straight N.
rightwards double arrow   P(m) : straight m cubed space plus space left parenthesis straight m space plus space 1 right parenthesis cubed space plus space left parenthesis straight m space plus space 2 right parenthesis cubed is divisible by 9.
rightwards double arrow   <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>                           ...(i)
III.   For n = m + 1,                           
       straight P left parenthesis straight m plus 1 right parenthesis space colon space left parenthesis straight m plus 1 right parenthesis cubed space plus space left parenthesis straight m plus 2 right parenthesis cubed space plus space left parenthesis straight m plus 3 right parenthesis cubed is divisible by 9.
       Now, from (i),
     
       space space straight m cubed space plus space left parenthesis straight m plus 1 right parenthesis cubed space plus space left parenthesis straight m plus 2 right parenthesis cubed space equals space 9 straight k      
rightwards double arrow     left parenthesis straight m plus 1 right parenthesis cubed plus left parenthesis straight m plus 2 right parenthesis cubed space equals space 9 straight k space minus space straight m cubed space space rightwards double arrow space left parenthesis straight m plus 1 right parenthesis cubed space plus space left parenthesis straight m plus 2 right parenthesis cubed space plus space left parenthesis straight m space plus space 3 right parenthesis cubed space equals space 9 straight k space minus space straight m cubed space plus space left parenthesis straight m plus 3 right parenthesis cubed
         negative 9 straight k minus straight m cubed plus straight m cubed plus 27 space plus space 3 straight m squared left parenthesis 3 right parenthesis space plus space 3 space left parenthesis 9 right parenthesis space straight m space equals space 9 straight k plus 27 plus 9 straight m squared plus 27 straight m space equals space 9 left parenthesis straight k plus 3 plus straight m squared plus 3 straight m right parenthesis
         where straight k space element of space straight Z comma space space space straight m space element of space straight N
rightwards double arrow     <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 9
rightwards double arrow       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 space element of space straight N.




More Chapters from Sets