Question

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

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

Solution

Let P(n): $1 half plus space 1 fourth plus 1 over 8 plus....... plus 1 over 2 to the power of straight n space equals space 1 minus space 1 over 2 to the power of straight n$
I.    For n = 1,
     P(1) : $1 half space equals space 1 space minus space 1 over 2 to the power of 1 rightwards double arrow space 1 half space equals space 1 space minus space 1 half space space rightwards double arrow space 1 half space equals space 1 half$
∴    P(1) is true.
II.   Let the statement be true
       for n = m, $straight m space element of space straight N$

∴ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ .... (i)

III. For n = m + 1,
 P(m + 1): $1 half plus 1 fourth plus 1 over 8 plus......... plus 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 space minus space 1 over 2 to the power of straight m plus 1 end exponent$
or $space space 1 half space plus space 1 fourth space plus space 1 over 8 space plus space......... space plus space 1 over 2 to the power of straight m plus 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 space minus 1 over 2 to the power of straight m plus 1 end exponent$
 From (i),
 $1 half space plus space 1 fourth space plus space 1 over 8 space plus space.......... space plus space 1 over 2 to the power of straight m space equals space 1 space minus space 1 over 2 to the power of straight m$

∴   $straight P left parenthesis straight m plus 1 right parenthesis colon space 1 space minus space 1 over 2 to the power of straight m space plus space 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 space minus space 1 over 2 to the power of straight m plus 1 end exponent$
$rightwards double arrow space 1 minus open parentheses 1 over 2 to the power of straight m minus 1 over 2 to the power of straight m plus 1 end exponent close parentheses space equals space 1 space minus space 1 over 2 to the power of straight m plus 1 end exponent space rightwards double arrow space 1 minus space open parentheses fraction numerator 2 minus 1 over denominator 2 to the power of straight m plus 1 end exponent end fraction close parentheses space equals space 1 minus space 1 over 2 to the power of straight m plus 1 end exponent space rightwards double arrow space 1 minus 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 minus space 1 over 2 to the power of straight m plus 1 end exponent$
      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 $straight n element of straight N$ .

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