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

$1 plus 3 plus 3 squared plus....... space plus 3 to the power of straight n minus 1 end exponent space equals space fraction numerator 3 to the power of straight n minus 1 over denominator 2 end fraction$

Solution

Let P(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. For n = 1,
P(1) : 1 $equals fraction numerator 3 to the power of 1 minus 1 over denominator 2 end fraction space rightwards double arrow space 1 space equals space 2 over 2 space rightwards double arrow space space space 1 space equals space 1$

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

∴ P(m) : $1 plus 3 plus 3 squared plus space........... space plus space 3 to the power of straight m minus 1 end exponent space equals space fraction numerator 3 to the power of straight m minus 1 over denominator 2 end fraction$ ... (i)

III. For n = m + 1,
 P(m + 1) : $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
or $straight P left parenthesis straight m plus 1 right parenthesis space colon space 1 plus 3 plus 3 squared plus.......... plus 3 to the power of straight m minus 1 end exponent space plus 3 to the power of straight m equals space fraction numerator 3 to the power of straight m minus 1 end exponent minus 1 over denominator 2 end fraction$
From (i), $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

∴ $straight P left parenthesis straight m space plus 1 right parenthesis space colon space open parentheses fraction numerator 3 to the power of straight m minus 1 over denominator 2 end fraction close parentheses space plus space 3 to the power of straight m space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction$
$open parentheses fraction numerator 3 to the power of straight m minus 1 space plus space 2. space 3 to the power of straight m over denominator 2 end fraction close parentheses space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction rightwards double arrow space space fraction numerator 3.3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction space space rightwards double arrow space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction$
which is true.

∴ P(m + 1) is true.

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

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Prove the following by using the principle of mathematical induction for all :

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