Question

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

$1 plus 3 plus 5 plus........... space plus space left parenthesis 2 straight n minus 1 right parenthesis space equals space straight n squared$

Solution

Let P(n): $1 plus 3 plus 5 plus............. plus left parenthesis 2 straight n minus 1 right parenthesis space equals space straight n squared$
I.         For n = 1,
           P(1) : 1 = $left parenthesis 1 right parenthesis squared rightwards double arrow space 1 space equals space 1$
∴         P(1) is true
II.        Let the statement be true for n = m, $straight m element of straight N$
$rightwards double arrow$      P(m) : 1 + 3 + 5 + .................... + (2m - 1) = m²                    ...(i)
III.      For n = m + 1,
         P(m + 1) : 1 + 3 + 5 + .......... + [2 (m+1) - 1] = (m + 1)²
or      1 + 3 + 5 + ........... + (2m - 1) + (2m + 1) = (m + 1)²
        From (i),
         $1 plus 3 plus 5 plus.............. space plus space left parenthesis 2 straight m space minus space 1 right parenthesis space equals space straight m squared$

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

∴ P (m + 1) is true.

∴ P(m) is true. $rightwards double arrow space straight P left parenthesis straight m space plus space 1 right parenthesis$ is true.
Hence, by the principal of mathematical induction, P(n) is true for all $space space straight n element of straight N.$

For Daily Free Study Material Join wiredfaculty

Sets

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

$1 plus 3 plus 5 plus........... space plus space left parenthesis 2 straight n minus 1 right parenthesis space equals space straight n squared$

Some More Questions From Sets Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction