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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.

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

The given sequence is $space space open curly brackets straight t subscript straight n close curly brackets$ defined by $space space space space space space straight t subscript straight n equals space 2 to the power of straight n space plus space 3 straight n$

                           $straight t subscript 1 space equals space 2 to the power of 1 space plus space 3. space 1 space equals space 2 to the power of 1 space plus space 3$

                           $straight t subscript 2 space equals space 2 squared space plus space 3. space 2 space equals space 2 squared plus 6$

                            $straight t subscript 3 space equals space 2 cubed space plus space 3. space 3 space equals space 2 cubed space plus space 9$

Let $space space straight S subscript straight n$ denote the sum of n terms

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

 = $left parenthesis 2 to the power of 1 plus 3 right parenthesis space plus space left parenthesis 2 squared plus 6 right parenthesis space plus space left parenthesis 2 cubed plus 9 right parenthesis space plus space........... space plus space left parenthesis 2 to the power of straight n plus 3 straight n right parenthesis$

 $equals left parenthesis 2 to the power of 1 plus 2 squared plus 2 cubed plus...... plus 2 to the power of straight n right parenthesis space plus space left parenthesis 3 plus 6 plus 9 plus....... plus 3 straight n right parenthesis$

Since the series in the first bracket is a G.P. series with 1st term 2, common ratio 2 and the series in the second bracket is an A.P. with 1st term 3, common difference 3.

∴ $straight S subscript straight n space equals space fraction numerator 2 left parenthesis 2 to the power of straight n minus 1 right parenthesis over denominator 2 minus 1 end fraction space plus space straight n over 2 left square bracket 3 plus 3 straight n right square bracket space equals space 2 to the power of straight n plus 1 end exponent minus 2 space plus space fraction numerator 3 straight n over denominator 2 end fraction left parenthesis straight n plus 1 right parenthesis$