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How many terms of the sequence

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

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82.

Sum the following series:

7 + 77 + 777 + ......... to n terms

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83.

Sum the following series:

0.5 + 0.55 + 0.555.......... to n terms

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84.
Given a G.P. with a = 729 and 7th term 64, determine S₇.

Here a = 729, Let r be the common ratio

$space space straight t subscript 7 space equals space 64 space space space space space space space space space space space space space space space rightwards double arrow space space space ar to the power of 7 minus 1 end exponent space equals space 64 space rightwards double arrow space 729. space straight r to the power of 6 space equals space 64$

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

When $space space straight r space equals space 2 over 3 comma$ then

$straight S subscript 7 space equals space straight a fraction numerator left parenthesis 1 minus straight r to the power of 7 right parenthesis over denominator 1 minus straight r end fraction space equals space fraction numerator 729 open square brackets 1 minus open parentheses begin display style 2 over 3 end style close parentheses to the power of 7 close square brackets over denominator 1 minus begin display style 2 over 3 end style end fraction space space equals space 2187 open square brackets 1 minus 128 over 2187 close square brackets space equals space 2187 space open square brackets fraction numerator 2187 minus 128 over denominator 2187 end fraction close square brackets space equals space 2059$

When $straight r space equals space fraction numerator negative 2 over denominator 3 end fraction comma$ then

$straight S subscript 7 space equals space fraction numerator straight a left parenthesis 1 minus straight r to the power of 7 right parenthesis over denominator 1 minus straight r end fraction space equals space fraction numerator 729 open square brackets 1 minus open parentheses begin display style fraction numerator negative 2 over denominator 3 end fraction end style close parentheses to the power of 7 close square brackets over denominator 1 minus open parentheses begin display style fraction numerator negative 2 over denominator 3 end fraction end style close parentheses end fraction space equals space 2187 over 5 open square brackets 1 plus 128 over 2187 close square brackets space equals space fraction numerator 2187 plus 128 over denominator 5 end fraction space equals space 2315 over 5 space equals space 463$

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85.

Determine the number of terms in a GP., if t₁ = 3, t_n = 96 and S_n = 189.

243 Views

86.

Find the least value of n for which the sum 1 + 3 + 3²+ ...... to n terms is greater than 7000.

321 Views

87.

The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.

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88.

If S₁, S₂ and S₃ be respectively the sums of n, 2n and 3n terms of a GP., prove that S₁ (S₃ – S₂) = (S₂ – S₁)²

1051 Views

Long Answer Type

89.

Let S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. Prove that $straight P squared straight R to the power of straight n space equals space straight S to the power of straight n.$

131 Views

Short Answer Type

90.

Evaluate:

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