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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.

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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

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

Let $space space straight S subscript straight n$ denote the sum of first n terms of the given sequence.

∴            $straight S subscript straight n space equals space open parentheses straight x plus 1 over straight x close parentheses squared plus open parentheses straight x squared plus 1 over straight x squared close parentheses squared plus open parentheses straight x plus 1 over straight x cubed close parentheses squared plus......... plus open parentheses straight x to the power of straight n plus 1 over straight x to the power of straight n close parentheses squared$

                 $equals open parentheses straight x squared plus 1 over straight x squared plus 2 close parentheses space plus space open parentheses straight x to the power of 4 plus 1 over straight x to the power of 4 plus 2 close parentheses plus open parentheses straight x to the power of 6 plus 1 over straight x to the power of 6 plus 2 close parentheses plus..... plus open parentheses straight x to the power of 2 straight n end exponent plus 1 over straight x to the power of 2 straight n end exponent plus 2 close parentheses$

                 = $left square bracket straight x squared plus straight x to the power of 4 plus straight x to the power of 6 plus......... straight x to the power of 2 straight n end exponent right square bracket space plus space open square brackets 1 over straight x squared plus 1 over straight x to the power of 4 plus 1 over straight x to the power of 6 plus...... plus 1 over straight x to the power of 2 straight n end exponent close square brackets plus 2 straight n$

                = $fraction numerator straight x squared left square bracket left parenthesis straight x squared right parenthesis to the power of straight n minus 1 right square bracket over denominator straight x squared minus 1 end fraction plus fraction numerator begin display style 1 over straight x squared end style open square brackets 1 minus open parentheses begin display style 1 over straight x squared end style close parentheses to the power of straight n close square brackets over denominator 1 minus begin display style 1 over straight x squared end style end fraction plus 2 straight n space equals space straight x squared open parentheses fraction numerator straight x to the power of 2 straight n minus 1 end exponent over denominator straight x squared minus 1 end fraction close parentheses plus fraction numerator left parenthesis straight x to the power of 2 straight n end exponent minus 1 right parenthesis over denominator straight x to the power of 2 straight n end exponent left parenthesis straight x squared minus 1 right parenthesis end fraction plus 2 straight n$

                                                                         $open curly brackets because space space In space straight G. straight P. comma space space straight S subscript straight n space equals space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction close curly brackets$

               $equals open parentheses fraction numerator straight x to the power of 2 straight n end exponent minus 1 over denominator straight x squared minus 1 end fraction close parentheses open parentheses straight x squared plus 1 over straight x to the power of 2 straight n end exponent close parentheses plus 2 straight n$

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