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

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52.

If a, b, c are in A.P., prove that $fraction numerator 1 over denominator square root of straight b plus square root of straight c end fraction comma space fraction numerator 1 over denominator square root of straight c plus square root of straight a end fraction comma space fraction numerator 1 over denominator square root of straight a plus square root of straight b space end root end fraction$ are also in A.P.

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53.

If $straight a squared comma space straight b squared comma space straight c squared$ are in A.P., then prove that $fraction numerator straight a over denominator straight b plus straight c end fraction comma space fraction numerator straight b over denominator straight c plus straight a end fraction comma space fraction numerator straight c over denominator straight a plus straight b end fraction$ are in A.P.

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54.

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55.

If $space space left parenthesis straight b minus straight c right parenthesis squared comma space left parenthesis straight c minus straight a right parenthesis squared comma space left parenthesis straight a minus straight b right parenthesis squared$ are in A.P., prove that $fraction numerator 1 over denominator straight b minus straight c end fraction comma space fraction numerator 1 over denominator straight c minus straight a end fraction comma space fraction numerator 1 over denominator straight a minus straight b end fraction$ are in A.P.

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56.

If a, b, c are in A.P. then prove that $left parenthesis straight a minus straight c right parenthesis squared space equals space 4 left parenthesis straight b squared minus ac right parenthesis$

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57.
If a, b, c are in A.P. then prove that

$straight a cubed plus 4 straight b cubed plus straight c cubed space equals space 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis$

Since a, b, c are in A.P.

∴ b - a = c - b or 2b = a + c or b = $fraction numerator straight a plus straight c over denominator 2 end fraction$ ...(i)

$straight a cubed plus 4 straight b cubed plus straight c cubed equals 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis$

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

if $straight a cubed plus fraction numerator 4 left parenthesis straight a plus straight c right parenthesis cubed over denominator 8 end fraction plus straight c cubed space equals space fraction numerator 3 left parenthesis straight a cubed plus ac squared plus ca squared plus straight c cubed right parenthesis over denominator 2 end fraction$

if $fraction numerator 2 straight a cubed plus straight a cubed plus straight c cubed plus 3 straight a squared straight c plus 3 ac squared plus 2 straight c cubed over denominator 2 end fraction space equals space fraction numerator 3 left parenthesis straight a cubed plus ac squared plus ca squared plus straight c cubed right parenthesis over denominator 2 end fraction$

if $3 straight a cubed plus 3 straight c cubed plus 3 straight a squared straight c plus 3 ac squared space equals space 3 straight a cubed plus 3 straight c cubed plus 3 straight a squared straight c plus 3 ac squared$ which is true

∴ $straight a cubed plus 4 straight b cubed plus straight c cubed space equals space 3 straight b left parenthesis straight a squared plus straight c squared right parenthesis$

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58.

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