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Sequences and Series

Short Answer Type

71.

Show that products of the corresponding terms of sequences a, ar, ar², .........ar ^{n – 1} and A, AR, AR², ........ A^{n – 1} form a GP. and find common ratio.

117 Views

72.
The first term of GP. is 1. The sum of 3rd and 5th term is 90. Find the common ratio of G.P.

Let a be first term and r be the common ratio of G.P.

Given a = 1, also, $straight t subscript 3 space plus space straight t subscript 5 space equals space 90$

$rightwards double arrow$ $ar squared space plus space ar to the power of 4 space equals space 90$

$rightwards double arrow$ $space space straight r squared plus straight r to the power of 4 space equals space 90$ (∵ a = 1)

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

$rightwards double arrow$ $space space straight r to the power of 4 plus 10 straight r squared minus 9 straight r squared minus 90 space equals space 0$ or $straight r squared left parenthesis straight r squared plus 10 right parenthesis space minus space 9 left parenthesis straight r squared plus 10 right parenthesis space equals space 0$

or $left parenthesis straight r squared minus 9 right parenthesis space left parenthesis straight r squared plus 10 right parenthesis space equals space 0$

$rightwards double arrow$ $straight r squared space minus space 9 space space equals 0$ $left parenthesis straight r squared plus 10 equals 0 space space is space impossible right parenthesis$

$rightwards double arrow$ $straight r space equals space plus-or-minus space 3$

118 Views

73.

Find the sum of the indicated terms of each of the following CP’s:

3, 6, 12, ........; 7 terms

82 Views

74.

Find the sum of the indicated terms of each of the following CP’s:

$square root of 7 comma space square root of 21 comma space 3 square root of 7 comma space........ semicolon space space straight n space terms$

115 Views

75.

Find the sum to indicated number of terms in each of the geometric progressions:

1, - a, $straight a squared comma space$ $negative straight a cubed comma space space space..... semicolon space$ n terms $left parenthesis straight a not equal to space minus 1 right parenthesis$

104 Views

76.

Find the sum to indicated number of terms in each of the geometric progressions:

$open parentheses 1 fifth plus 2 over 5 squared plus 3 over 5 cubed close parentheses comma space space open parentheses 1 over 5 to the power of 4 plus 2 over 5 to the power of 5 plus 3 over 5 to the power of 6 close parentheses comma space open parentheses 1 over 5 to the power of 7 plus 2 over 5 to the power of 8 plus 3 over 5 to the power of 9 close parentheses comma space...... semicolon 7$

114 Views

77.

Find the sum of n terms of sequence

$open parentheses straight x plus 1 over straight x close parentheses squared comma space space open parentheses straight x squared plus 1 over straight x squared close parentheses squared comma space space open parentheses straight x cubed plus 1 over straight x cubed close parentheses squared comma space space............$

109 Views

78.

Prove that the sum to n terms of the series

11 + 103 + 1005 +........ is $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

897 Views

79.

Find the sum to n terms of the sequence $space space open curly brackets straight t subscript straight n close curly brackets$ given by $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$