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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 $straight n space element of space straight N$ :

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

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13.

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

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

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14.

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

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

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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.

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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 $straight n space element of space straight N.$

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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.

Let P(n) : $10 to the power of 2 straight n minus 1 end exponent space plus space 1$ is divisible by 11

I. For n = 1,

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$rightwards double arrow$ 10¹ + 1 is divisible by 11 $rightwards double arrow$ 11 is divisible by 11

∴ P(1) is true

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

∴         P(m) : $10 to the power of 2 straight m minus 1 end exponent space plus space 1$ is divisible by 11.

$rightwards double arrow$ $10 to the power of 2 straight m minus 1 end exponent space plus space 1 space equals space 11 straight k comma space straight k space element of space straight Z$

$rightwards double arrow$        $10 to the power of 2 straight m minus 1 end exponent space equals space 11 straight k space minus space 1$                                          ...(i)

III.         For n = m + 1,

            $straight P left parenthesis straight m space plus space 1 right parenthesis space colon space space 10 to the power of 2 left parenthesis straight m plus 1 right parenthesis minus 1 end exponent plus 1$ is divisible by 11.       ...(ii)

Now, $10 to the power of 2 left parenthesis straight m plus 1 right parenthesis minus 1 end exponent plus 1 space equals space 10 to the power of 2 straight m minus 1 end exponent. space 10 squared space plus space 1 space equals space left parenthesis 11 straight k space minus space 1 right parenthesis. space 10 squared space plus space 1$        [By (i)]

            = $1100 straight k space minus space 100 space plus space 1 space equals space 1100 straight k space minus space 99 space equals space 11 space left parenthesis 100 straight k space minus space 9 right parenthesis space equals space 11 straight k apostrophe$

                                                                          where k' = 100k - $9 space element of space straight Z$

∴ $10 to the power of 2 left parenthesis straight m plus 1 right parenthesis minus 1 end exponent space plus space 1$ is divisible by 11

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

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

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

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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.