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Principle of Mathematical Induction

Long Answer Type

11.

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

a + (a + d) + (a + 2d) + ...........+ [a + (n - 1)d] = $straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket$

265 Views

12.

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

$open parentheses 1 plus 1 over 1 close parentheses open parentheses 1 plus 1 half close parentheses open parentheses 1 plus 1 third close parentheses space........ open parentheses 1 plus 1 over straight n close parentheses space equals space left parenthesis straight n space plus space 1 right parenthesis$

137 Views

13.

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

$space space 1 plus 2 plus 3 plus........ plus straight n less than 1 over 8 left parenthesis 2 straight n plus 1 right parenthesis squared$

124 Views

14.

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

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

265 Views

15.

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

n (n + 1) (n + 5) is a multiple of 3.

173 Views

16.

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

$41 to the power of straight n space minus space 14 to the power of straight n$ is a multiple of 27 for all $straight n space element of space straight N.$

171 Views

17.

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

$10 to the power of 2 straight n minus 1 end exponent space plus space 1$ is divisible by 11.

144 Views

18.
Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N.$

$3 to the power of 2 straight n plus 2 end exponent minus 8 straight n minus 9$ is divisible by 8.

Let $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ is divisible by 8.

I. For n = 1,

 P(1) : $3 to the power of 2 left parenthesis 1 right parenthesis plus 2 end exponent space minus space 8 space left parenthesis 1 right parenthesis space minus space 9$ is divisible by 8

$rightwards double arrow$ $3 to the power of 4 minus 8 minus 9$ is divisible by 8 $rightwards double arrow$ 81 - 17 is divisible by 8 $rightwards double arrow$ 64 is divisible by 8

 which is true

∴ P(n) is true for n = 1

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

∴ $straight P left parenthesis straight m right parenthesis space colon space 3 to the power of 2 straight m plus 2 end exponent space minus space 8 straight m space minus space 9$ is divisible by 8

$rightwards double arrow$ $space space space space 3 to the power of 2 straight m plus 2 end exponent space minus space 8 straight m space minus space 9 space equals space 8 straight k comma space straight k space element of space straight Z space rightwards double arrow space 3 to the power of 2 straight m plus 2 end exponent space equals space 8 straight k space plus space 8 straight m space plus space 9$ ...(i)

III. For n = m + 1,

 $space space space straight P left parenthesis straight m plus 1 right parenthesis colon space 3 to the power of 2 left parenthesis straight m plus 1 right parenthesis plus 2 end exponent space minus space 8 left parenthesis straight m space plus space 1 right parenthesis space minus space 9$ is divisible by 8

Now, $3 to the power of 2 left parenthesis straight m plus 1 right parenthesis plus 2 end exponent space minus space 8 left parenthesis straight m plus 1 right parenthesis minus 9 space equals space 3 to the power of 2 straight m plus 2 end exponent space. space 3 squared minus 8 straight m minus 8 minus 9$

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

= 72k + 72m + 81 - 8m - 17 = 72k + 64m + 64 = 8(9k + 8m + 8), $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

= 8k' where k' = 9k + 8m + $8 space element of space straight Z$

∴ $3 to the power of 2 left parenthesis straight m plus 1 right parenthesis plus 2 end exponent space minus space 8 left parenthesis straight m space plus space 1 right parenthesis space minus space 9$ is divisible by 8.

$rightwards double arrow$ P(m + 1) is true.

∴ P(m) is true $rightwards double arrow$ P (m + 1) is true.

Hence, by the principle of mathematical induction, P(n) is true for all $straight n space element of space straight N.$