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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] = $straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket$

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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.

160 Views

19.
Prove by mathematical induction that sum of cubes of three consecutive natural numbers is divisible by 9.

Let n, n+1, n+2 be three consecutive natural numbers.

Let P(n): $straight n cubed space plus space left parenthesis straight n plus 1 right parenthesis cubed space plus space left parenthesis straight n plus 2 right parenthesis cubed$ is divisible by 9.

I. For n = 1,

$straight P left parenthesis 1 right parenthesis space colon space 1 cubed space plus space 2 cubed space plus space 3 cubed$ is divisible by 9

$rightwards double arrow$ 1 + 8 + 27 is divisible by 9 $rightwards double arrow$ 36 is divisible by 9

which is true

∴ the statement is true for n = 1.

II. Suppose the statement is true for n = m, $straight m space element of space straight N.$

$rightwards double arrow$ P(m) : $straight m cubed space plus space left parenthesis straight m space plus space 1 right parenthesis cubed space plus space left parenthesis straight m space plus space 2 right parenthesis cubed$ is divisible by 9.

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

III. For n = m + 1,

 $straight P left parenthesis straight m plus 1 right parenthesis space colon space left parenthesis straight m plus 1 right parenthesis cubed space plus space left parenthesis straight m plus 2 right parenthesis cubed space plus space left parenthesis straight m plus 3 right parenthesis cubed$ is divisible by 9.

 Now, from (i),

 $space space straight m cubed space plus space left parenthesis straight m plus 1 right parenthesis cubed space plus space left parenthesis straight m plus 2 right parenthesis cubed space equals space 9 straight k$

$rightwards double arrow$ $left parenthesis straight m plus 1 right parenthesis cubed plus left parenthesis straight m plus 2 right parenthesis cubed space equals space 9 straight k space minus space straight m cubed space space rightwards double arrow space left parenthesis straight m plus 1 right parenthesis cubed space plus space left parenthesis straight m plus 2 right parenthesis cubed space plus space left parenthesis straight m space plus space 3 right parenthesis cubed space equals space 9 straight k space minus space straight m cubed space plus space left parenthesis straight m plus 3 right parenthesis cubed$

 $negative 9 straight k minus straight m cubed plus straight m cubed plus 27 space plus space 3 straight m squared left parenthesis 3 right parenthesis space plus space 3 space left parenthesis 9 right parenthesis space straight m space equals space 9 straight k plus 27 plus 9 straight m squared plus 27 straight m space equals space 9 left parenthesis straight k plus 3 plus straight m squared plus 3 straight m right parenthesis$

 where $straight k space element of space straight Z comma space space space straight m space element of space straight N$

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

$rightwards double arrow$ 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 $straight n space element of space straight N.$