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

$space space fraction numerator 1 over denominator straight n plus 1 end fraction space plus space fraction numerator 1 over denominator straight n plus 2 end fraction space plus space.......... space plus space fraction numerator 1 over denominator 2 straight n end fraction greater than 13 over 24$

Solution

Let $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
I. For n = 2(note this step, n>1)
 $straight P left parenthesis 2 right parenthesis colon space space fraction numerator 1 over denominator 2 plus 1 end fraction plus fraction numerator 1 over denominator 2 plus 2 end fraction greater than 13 over 24$
$space space rightwards double arrow space space space 1 third space plus space 1 fourth greater than 13 over 24 space space rightwards double arrow space 7 over 12 greater than 13 over 24 space space space rightwards double arrow space 14 over 24 greater than 13 over 24$
 which is true

∴ P(n) is true for n = 2
II. Suppose the statement is true for n = m, $straight m space element of space straight N comma space space straight m greater than 1.$
$rightwards double arrow$ $space space straight P left parenthesis straight m right parenthesis space equals space fraction numerator 1 over denominator straight m plus 1 end fraction plus fraction numerator 1 over denominator straight m plus 2 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m end fraction greater than 13 over 24$ .... (i)
III. For n = 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 $fraction numerator 1 over denominator straight m plus 2 end fraction space plus space fraction numerator 1 over denominator straight m plus 3 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24$
or $space space fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus..... plus fraction numerator 1 over denominator 2 straight m end fraction plus fraction numerator 1 over denominator 2 straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24$
 Adding $fraction numerator 1 over denominator 2 straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction$ on both sides of (i), we get

 $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24 plus space fraction numerator 2 straight m plus 2 plus 2 straight m plus 1 minus 4 straight m minus 2 over denominator 2 space left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis straight m plus 1 right parenthesis end fraction$
$rightwards double arrow$ $fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus......... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24 plus fraction numerator 1 over denominator 2 left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction$
 But, $fraction numerator 1 over denominator 2 space left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction greater than 0$

∴ $space space fraction numerator 1 over denominator straight m plus 2 end fraction space plus space fraction numerator 1 over denominator straight m plus 3 end fraction plus......... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24$

∴ P (m + 1) is true
∴ P(m) is true $rightwards double arrow$ P(m + 1) is true
Hence, P(n) is true for all $straight n element of straight N comma space straight n greater than 1.$

Delhi University

NCERT Book Store

NCERT Sample Papers

Entrance Exams Preparation

Company

Term of Use

Privacy Policy

NCERT Solutions for Class-7th

NCERT Solutions for Class-8th

NCERT Solutions for Class-9th

NCERT Solutions for Class-10th

NCERT Solutions for Class-11th

NCERT Solutions for Class-12th

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

Delhi University

NCERT Book Store

NCERT Sample Papers

Entrance Exams Preparation