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Principle Of Mathematical Induction

Question

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The 4^th term in the expansion of $open parentheses ax plus 1 over straight x close parentheses to the power of straight n$ is $<pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ Find the values of a and n.

Solution

$4 to the power of th$ term in the expansion of $open parentheses ax plus 1 over straight x close parentheses to the power of straight n$ = $5 over 2$

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<pre>uncaught exception: <b>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</b><br /><br />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<br />#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>

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space space straight C presuperscript straight n subscript 3 space left parenthesis ax right parenthesis to the power of straight n minus 3 end exponent. space open parentheses 1 over straight x close parentheses cubed space equals space 5 over 2

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straight C presuperscript straight n subscript 3 space straight a to the power of straight n minus 3 end exponent. space straight x to the power of straight n minus 3 end exponent space cross times 1 over straight x cubed equals 5 over 2

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straight C presuperscript straight n subscript 3 space straight a to the power of straight n minus 3 end exponent. space straight x to the power of straight n minus 6 end exponent space equals space 5 over 2

...(i)
The R.H.S. is independent of x

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straight x to the power of straight n minus 6 end exponent space equals space straight x to the power of 0

or n - 6 = 0
∴ n = 6
Also, from (i),

straight C presuperscript 6 subscript 3 space straight a to the power of 6 minus 3 end exponent. space straight x to the power of 0 space equals space 5 over 2

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fraction numerator 6 factorial over denominator 3 factorial space 3 factorial end fraction straight a cubed space equals space 5 over 2

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20 space straight a cubed space equals space 5 over 2

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straight a cubed space equals space fraction numerator 5 over denominator 2 cross times 20 end fraction equals 1 over 8 equals open parentheses 1 half close parentheses cubed

∴

straight a equals 1 half

Hence, n = 6 and

straight a equals 1 half.

Some More Questions From Principle of Mathematical Induction Chapter

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

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