Permutations and Combinations

Question

The interior angles of a polygon are in A.P. The smallest angle is 120° and the common difference is 5°. Find the number of sides of the polygon.

Solution

Let the number of sides of the polygon be n. The interior angles from an A.P. Let a be the smallest angle and d be the common difference.
∴ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Since the sum of all the interior angles of a polygon with n sides is $left parenthesis straight n minus 2 right parenthesis space 180 degree$
∴ $straight S subscript straight n space equals space left parenthesis straight n minus 2 right parenthesis cross times 180$
$rightwards double arrow$ $space space space straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket space equals space 180 straight n minus 360$ $rightwards double arrow$ $straight n over 2 left square bracket 2 left parenthesis 120 right parenthesis space plus space left parenthesis straight n minus 1 right parenthesis 5 right square bracket space equals space 180 straight n space minus space 360$
$rightwards double arrow$ $120 straight n space plus space fraction numerator 5 straight n squared minus 5 straight n over denominator 2 end fraction space equals space 180 straight n space minus space 360$ $rightwards double arrow$ $space space space 240 straight n space plus space 5 straight n squared minus 5 straight n space equals space 360 straight n space minus space 720$
$rightwards double arrow$ $5 straight n squared minus 125 straight n plus 720 space equals space 0$ $rightwards double arrow$ $straight n squared minus 25 straight n plus 144 equals 0$
$rightwards double arrow$ $straight n squared minus 9 straight n minus 16 straight n plus 144 space equals space 0$ $rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $space space left parenthesis straight n minus 9 right parenthesis space left parenthesis straight n minus 16 right parenthesis space equals space 0$ $rightwards double arrow$ n = 9, 16
But if n = 16, then $straight T subscript 16 space equals space straight a space plus space 15 straight d space equals space 120 degree space plus space 15 left parenthesis 5 degree right parenthesis space equals space 195 degree comma$ which is impossible.